I took the missus to Budapest for her birthday a couple of months ago, to be honest I reckon it has everything you need, certainly more than you can get through in a few days.
Plenty of museums, galleries and such, you can get into a few of them for ‘free’ by upgrading to a tourist travel card or get discounts for a lot of other stuff. The opera/ballet is well worth a visit and VERY cheap in comparison to what you get here. We went to a matinee performance of the Nutcracker the other day sitting in seats well off to the side of the stalls, the same money (£30 each) got us premium box seats at an evening performance. I also know of some good apartments right by the Opera house (central Pest): http://www.operaresidence.hu/
It’s a cheap place to visit, if I’m honest the main cost will be the travel, Jet2 certainly like to heap on the extras but once you’re there it’s comically cheap if you avoid the tourist traps. As far as nightlife goes, ruin bars are the way to go. They sound hipster-ish and likely to be pretentious tourist traps but honestly are great. We went to Szimpla-Kert (the original) and all in I don’t think we spent more than £20 over the evening!