According to the Castle Windsor documentation you can implement your own custom scope. You have to implement the Castle.MicroKernel.Lifestyle.Scoped.IScopeAccessor interface.
You then specify your scope accessor when registering your component:
Container.Register(Component.For<MyScopedComponent>().LifestyleScoped<OwinWebRequestScopeAccessor >());
The class OwinWebRequestScopeAccessor implements Castle.Windsor's IScopeAccessor:
using Castle.MicroKernel.Context;
using Castle.MicroKernel.Lifestyle.Scoped;
using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading;
using System.Threading.Tasks;
namespace Web.Api.Host
{
    public class OwinWebRequestScopeAccessor : IScopeAccessor
    {
        public void Dispose()
        {
            var scope = PerWebRequestLifestyleOwinMiddleware.YieldScope();
            if (scope != null)
            {
                scope.Dispose();
            }
        }
        public ILifetimeScope GetScope(CreationContext context)
        {
            return PerWebRequestLifestyleOwinMiddleware.GetScope();
        }
    }
}
As you can see OwinWebRequestScopeAccessor delegates the calls to GetScope and Dispose to PerWebRequestLifestyleOwinMiddleware. 
The class PerWebRequestLifestyleOwinMiddleware is the OWIN counter part of Castle Windsor's ASP.NET IHttpModule PerWebRequestLifestyleModule.
This is the PerWebRequestLifestyleOwinMiddleware class:
using Castle.MicroKernel;
using Castle.MicroKernel.Lifestyle.Scoped;
using Owin;
using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;
namespace Web.Api.Host
{
    using AppFunc = Func<System.Collections.Generic.IDictionary<string, object>, System.Threading.Tasks.Task>;
    public class PerWebRequestLifestyleOwinMiddleware
    {
        private readonly AppFunc _next;
        private const string c_key = "castle.per-web-request-lifestyle-cache";
        private static bool _initialized;
        public PerWebRequestLifestyleOwinMiddleware(AppFunc next)
        {
            _next = next;
        }
        public async Task Invoke(IDictionary<string, object> environment)
        {
            var requestContext = OwinRequestScopeContext.Current;
            _initialized = true;
            try
            {
                await _next(environment);
            }
            finally
            {
                var scope = GetScope(requestContext, createIfNotPresent: false);
                if (scope != null)
                {
                    scope.Dispose();
                }
                requestContext.EndRequest();
            }
        }
        internal static ILifetimeScope GetScope()
        {
            EnsureInitialized();
            var context = OwinRequestScopeContext.Current;
            if (context == null)
            {
                throw new InvalidOperationException(typeof(OwinRequestScopeContext).FullName +".Current is null. " +
                    typeof(PerWebRequestLifestyleOwinMiddleware).FullName +" can only be used with OWIN.");
            }
            return GetScope(context, createIfNotPresent: true);
        }
        /// <summary>
        /// Returns current request's scope and detaches it from the request 
        /// context. Does not throw if scope or context not present. To be 
        /// used for disposing of the context.
        /// </summary>
        /// <returns></returns>
        internal static ILifetimeScope YieldScope()
        {
            var context = OwinRequestScopeContext.Current;
            if (context == null)
            {
                return null;
            }
            var scope = GetScope(context, createIfNotPresent: false);
            if (scope != null)
            {
                context.Items.Remove(c_key);
            }
            return scope;
        }
        private static void EnsureInitialized()
        {
            if (_initialized)
            {
                return;
            }
            throw new ComponentResolutionException("Looks like you forgot to register the OWIN middleware " + typeof(PerWebRequestLifestyleOwinMiddleware).FullName);
        }
        private static ILifetimeScope GetScope(IOwinRequestScopeContext context, bool createIfNotPresent)
        {
            ILifetimeScope candidates = null;
            if (context.Items.ContainsKey(c_key))
            {
                candidates = (ILifetimeScope)context.Items[c_key];
            }
            else if (createIfNotPresent)
            {
                candidates = new DefaultLifetimeScope(new ScopeCache());
                context.Items[c_key] = candidates;
            }
            return candidates;
        }
    }
    public static class AppBuilderPerWebRequestLifestyleOwinMiddlewareExtensions
    {
        /// <summary>
        /// Use <see cref="PerWebRequestLifestyleOwinMiddleware"/>.
        /// </summary>
        /// <param name="app">Owin app.</param>
        /// <returns></returns>
        public static IAppBuilder UsePerWebRequestLifestyleOwinMiddleware(this IAppBuilder app)
        {
            return app.Use(typeof(PerWebRequestLifestyleOwinMiddleware));
        }
    }
}
Castle Windsor's ASP.NET IHttpModule PerWebRequestLifestyleModule utilizes HttpContext.Current for storing the Castle Windsor ILifetimeScope on a per-web-request basis. PerWebRequestLifestyleOwinMiddleware class uses OwinRequestScopeContext.Current. This is based on the idea of Yoshifumi Kawai.
The implementation of OwinRequestScopeContext below is my lightweight implementation of Yoshifumi Kawai's original OwinRequestScopeContext. 
Note: if you don't want this lightweight implementation you can use Yoshifumi Kawai's excellent original implementation by running this command in the NuGet Package Manager Console:
PM> Install-Package OwinRequestScopeContext
Lightweight implementation of OwinRequestScopeContext:
using System;
using System.Collections.Concurrent;
using System.Collections.Generic;
using System.Linq;
using System.Runtime.Remoting.Messaging;
using System.Text;
using System.Threading.Tasks;
namespace Web.Api.Host
{
    public interface IOwinRequestScopeContext
    {
        IDictionary<string, object> Items { get; }
        DateTime Timestamp { get; }
        void EndRequest();
    }
    public class OwinRequestScopeContext : IOwinRequestScopeContext
    {
        const string c_callContextKey = "owin.reqscopecontext";
        private readonly DateTime _utcTimestamp = DateTime.UtcNow;
        private ConcurrentDictionary<string, object> _items = new ConcurrentDictionary<string, object>();
        /// <summary>
        /// Gets or sets the <see cref="IOwinRequestScopeContext"/> object 
        /// for the current HTTP request.
        /// </summary>
        public static IOwinRequestScopeContext Current
        {
            get
            {
                var requestContext = CallContext.LogicalGetData(c_callContextKey) as IOwinRequestScopeContext;
                if (requestContext == null)
                {
                    requestContext = new OwinRequestScopeContext();
                    CallContext.LogicalSetData(c_callContextKey, requestContext);
                }
                return requestContext;
            }
            set
            {
                CallContext.LogicalSetData(c_callContextKey, value);
            }
        }
        public void EndRequest()
        {
            CallContext.FreeNamedDataSlot(c_callContextKey);
        }
        public IDictionary<string, object> Items
        {
            get
            {
                return _items;
            }
        }
        public DateTime Timestamp
        {
            get
            {
                return _utcTimestamp.ToLocalTime();
            }
        }
    }
}
When you have all the pieces in place you can tie things up. In your OWIN startup class call the appBuilder.UsePerWebRequestLifestyleOwinMiddleware(); extension method to register the OWIN middle ware defined above. Do this before appBuilder.UseWebApi(config);:
using Owin;
using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;
using System.Web.Http;
using System.Diagnostics;
using Castle.Windsor;
using System.Web.Http.Dispatcher;
using System.Web.Http.Tracing;
namespace Web.Api.Host
{
    class Startup
    {
        private readonly IWindsorContainer _container;
        public Startup()
        {
            _container = new WindsorContainer().Install(new WindsorInstaller());
        }
        public void Configuration(IAppBuilder appBuilder)
        {
            var properties = new Microsoft.Owin.BuilderProperties.AppProperties(appBuilder.Properties);
            var token = properties.OnAppDisposing;
            if (token != System.Threading.CancellationToken.None)
            {
                token.Register(Close);
            }
            appBuilder.UsePerWebRequestLifestyleOwinMiddleware();
            //
            // Configure Web API for self-host. 
            //
            HttpConfiguration config = new HttpConfiguration();
            WebApiConfig.Register(config);
            appBuilder.UseWebApi(config);
        }
        public void Close()
        {
            if (_container != null)
                _container.Dispose();
        }
    }
}
The sample WindsorInstaller class shows how you can use the OWIN per-web-request scope:
using Castle.MicroKernel.Registration;
using Castle.MicroKernel.SubSystems.Configuration;
using Castle.Windsor;
using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;
namespace Web.Api.Host
{
    class WindsorInstaller : IWindsorInstaller
    {
        public void Install(IWindsorContainer container, IConfigurationStore store)
        {
            container.Register(Component
                .For<IPerWebRequestDependency>()
                .ImplementedBy<PerWebRequestDependency>()
                .LifestyleScoped<OwinWebRequestScopeAccessor>());
            container.Register(Component
                .For<Controllers.V1.TestController>()
                .LifeStyle.Transient);
        }
    }
}
The solution I have laid out above is based on the existing /src/Castle.Windsor/MicroKernel/Lifestyle/PerWebRequestLifestyleModule.cs and Yoshifumi Kawai'soriginal OwinRequestScopeContext.