I highly recommend doing a half day bike tour of the Pest side of the city, I think we used Budabike and they did tours starting outside the St Stephens Cathedral.
It is flat, it is quite slow because you have a mixed group, but it is a fantastic way to be in the city, get to lots of places in just a few hours and get some fantastic personal insights from the tour guides. And with the current political turmoil in Hungary it is particularly interesting to hear an insiders personal view of recent and more distant history.
Once you have done the tour it will give you lots of places you want to go back to on foot and see again at your leisure. Itt might sound a lame thing to do, but now I make going on a bike tour a top priority for any major city I visit, following my good experience on this a few years ago.