well some websites you can paste the link and it will download the videos in mp4 format or you can use some apps out there (ItubeX) i dont know if there is a windows version or if your on mac or windows so i cant really recommend any apps..
There are two steps for you to go, they are record video from YouTube and convert downloaded YouTube video for uploading to facebook. Of course, the most suitable format for uploading to facebook is MPEG4.
With this video download and record program, you will be able to record YouTube videos and save videos as mpeg4 format, then upload them to faceook with ease. You will get fast upload process in facebook friendly browser like latest version of IE and Firefox.