Solution 1

You can use YouTube Data API to access most of the information about the video, including duration:

<script type="text/javascript">
    function youtubeFeedCallback(json){
        document.write(json["data"]["duration"] + " second(s)");
    }
</script>
<script type="text/javascript" src="http://gdata.youtube.com/feeds/api/videos/4TSJhIZmL0A?v=2&alt=jsonc&callback=youtubeFeedCallback&prettyprint=true"></script>

Demo here

When using jQuery you can use $.getJSON() to make things easier.

Solution 2

Seems like YouTube JavaScript API v3 allows you to get the correct duration inside the onYouTubePlayerReady() event. All you need to do is pass &version=3 when calling swfobject.embedSWF() method.

Demo here