Christos,
If the user IDs are the same, you'll simply need to copy the #__jfbconnect_user_map table to the new site. However, you'll likely want (or even need) to use the same Facebook App ID and Secret Key on the new site. Facebook is moving to a unique user ID per app, which means that if you copy over the tables to the new site, but use a new App ID / Secret key, the user IDs that Facebook returns on authentication may be different than the ones stored in the database.
I hope that helps, but if you run into any issues, just let me know.
Alex