I'd go for the Castle Hill (mediaval old town). A cab ride would set you back ca. 15 euros each direction, 30-45 minutes are usually enough. so you'd have 3-4 hours to grab a decent lunch/dinner and enjoy the view. If you are thus inclined, the National Gallery and the Ludwig Museum are both there.
Definitely worth leaving the airport, as 6 hrs at BUD could be quite boring.