Budapest is a fun city... it has been about six years since I was there last... There are all sorts of hotels, just do an online search for the cheapest in the central area... most of the hotels there are nice. I stayed at the Taverna. It was right on the Vaci Utca (the walking street). It wasn't too extravagant.
If you have time try to spend a day at one of the hot baths... definately worth it. Also make sure, if you are a wine drinker, drink the local rose... it is very good. Also try the goose livers... it is a regional specialty. They are very good.
Never been to Bratislava so I can't help you there.
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