I am an American living in Budapest since July. I met some nice birders here through
www.birding.hu by emailing individuals who had posted in their forum and hoping they spoke english. I went on a few field trips with them and really was shown alot. But I have to say its not easy birding here for the following reasons.
Many of the best bird sites, such as the salt pans in the Kiskunsag or the fish ponds in the Hortobagy are not too close to Budapest and a car is required transit. There don't seem to be famous wildlife refuges which are walkable and well known as in the UK or eastern USA. The places aren't labelled for birding, with the exception of Hortobagy. It really takes a local guide to drive you to nondescript impoundments on private property but where birding is permited.
I was armed with the most detailed road atlas I could find and still we often drove on unmarked dirt roads across the dusty plain to arrive at an impoundment or salt flat which was not even on the map. I would not be able to find my way back to those great spots on my own.
2nd, while the Buda Hills are a great way to get away from town and hike. The area is forested hills and the habitat is quite common in Europe. It is to the south and east from Budapest that you find the dramatic & rare for Europe topography which hosts striking birds such as Roller, Beeeater, Hoopoe, Great Bustard, freshwater Terns, etc.
The Danube bend is also a forested hilly area. You may find Syrian Woodpecker, Rock Bunting and a few other species of interest, but mostly common forest birds of Europe. Also late Fall isn't the best time for Hunarian specialty birds. But you can still have a great time here.
Don't forget to look for Gulls along the Danube if you are here in winter. I am sorry I don't have better information. I have only been here since July. I have searched everwhere for info in english and talked to everyone I could but its not as well birded or documented (in english) a place as we are used to in the UK and USA.
If you are serious about more than just a birding stroll, I strongly recommend you contact one of the guide services and pay for a tour especially for the Puszta region. I have not done it myself yet, but I have 2 years to go....
If you have any specific questions, I will try to answer.
Marc