I think tha Borok haza is the most worthy thing to be visited in the old Buda. Forget about Fishermen bastion, Mathias church and all those gallerys in the King's palace! Go and taste some vines in Borok haza just some 100 meters away from these "must see places".
Favorite Dish: Tokaji. You must try it! But do me a favour: taste that one with 5 stars, not the cheapest one. Despite of not so little price you wont regret it.