I got back from Budapest two weeks ago after a long weekend with a couple of mates. AirBnB is definitely worth pursuing - it's what we did and were very happy with what we got!
Anyone and everyone will tell you to check out the Ruin Pubs. I'm not an avid beer drinker myself, but I did enjoy the several pubs we went to. There's plenty of usual tourist-type things to do, see the Basilica, museums, Palace etc. We stayed reasonably central and didn't need to use any public transport minus the taxi run from the airport and back!
Also, if you're changing currency, change most of it out there. Most sell rates I found on the UK high street were around 320-330 Forints per pound. Out there was at least 355 at every high street currency exchange. Likewise, try and change back whilst you're there.