How far back do you need to go in browser compatibility?  All of the modern browsers should support Object.getOwnPropertyNames().  Using your example, Object.getOwnPropertyNames([]) will return ["length"].
More info here:  https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Object/getOwnPropertyNames
Edit: Other examples:
- Object.getOwnPropertyNames([1, 2, 3]);returns- ["0", "1", "2", "length"]
 
- Object.getOwnPropertyNames(String);returns- ["prototype", "quote", "substring", "toLowerCase", "toUpperCase", "charAt", "charCodeAt", "contains", "indexOf", "lastIndexOf", "startsWith", "endsWith", "trim", "trimLeft", "trimRight", "toLocaleLowerCase", "toLocaleUpperCase", "localeCompare", "match", "search", "replace", "split", "substr", "concat", "slice", "fromCharCode", "length", "name", "arguments", "caller"]
 
Edit #2:  Okay, so seeing that you are looking for complete list of properties and methods, including inherited ones, I've borrowed from two other SO questions (linked below) and come up with a solution that appears to get you even closer:
var findProperties = function(obj) {
    var aPropertiesAndMethods = [];
    do {
        aPropertiesAndMethods = aPropertiesAndMethods.concat(Object.getOwnPropertyNames(obj));
    } while (obj = Object.getPrototypeOf(obj));
    for ( var a = 0; a < aPropertiesAndMethods.length; ++a) {
        for ( var b = a + 1; b < aPropertiesAndMethods.length; ++b) {
            if (aPropertiesAndMethods[a] === aPropertiesAndMethods[b]) {
                aPropertiesAndMethods.splice(a--, 1);
            }
        }
    }
    return aPropertiesAndMethods;
}
So if you use call findProperties([]), it returns ["length", "join", "reverse", "sort", "push", "pop", "shift", "unshift", "splice", "concat", "slice", "lastIndexOf", "indexOf", "forEach", "map", "reduce", "reduceRight", "filter", "some", "every", "iterator", "constructor", "toSource", "toString", "toLocaleString", "valueOf", "watch", "unwatch", "hasOwnProperty", "isPrototypeOf", "propertyIsEnumerable", "__defineGetter__", "__defineSetter__", "__lookupGetter__", "__lookupSetter__"]
Linked Questions
javascript inheritance, reflection and prototype chain walking?
How to merge two arrays in Javascript and de-duplicate items