You could do that using url parameters. For example, link to your site would be something like: www.mysite.com/something?video_url=<some youtube link>. Then, on you page you can have event handler ($(document).ready(function(){...})) which would check if there is a 'video_url' param present in the url, and if it is just show the popup with <some youtube link> loaded in it.
For example (getParameterByName(name) found here):
// a function that gets the value of query parameter
function getParameterByName(name) {
    name = name.replace(/[\[]/, "\\[").replace(/[\]]/, "\\]");
    var regex = new RegExp("[\\?&]" + name + "=([^&#]*)"),
    results = regex.exec(location.search);
    return results == null ? "" : decodeURIComponent(results[1].replace(/\+/g, " "));
}
// jquery used, but not really needed
$(document).ready(function() {
    var video_url = getParameterByName("video_url");
    // your function that shows the popup
    showPopup(video_url);
});