If you like WWII history you might enjoy the Hospital in the Rock up on Castle Hill--if you are facing St Mattias, turn around and walk straight across to the far end of the hill, look for the elevator that takes you down to the lower level. It's a hospital built into the natural caves, used during WWII and also the 1956 revolution, the guided tour (you can't do it alone, very cheap and excellent, informative guides) takes a bit over an hour and I found it fascinating, sorry I waited until my 4th trip to Budapest to see it.
If you go to Szentendre plan to spend 3-4 hours if you have a restaurant meal, I like a place on the main square called Elisabeth--sit outside, have some csirke paprikas and wine, people watch...I do believe I'll be there myself on May 10! Fridays are fun because they have a little marketplace going on, interesting crafts and food. My favorite museum is off the main square and is devoted to the quirky, enigmatic works of Hungary's first important female artist, ceramicist Margit Kovacs--it's small so doesn't take long, and her works are also scattered here and there on buildings in Budapest, notably the old post office on Castle hill. I'm a wee bit of a fan of hers, can you tell? :) https://www.tripadvisor.com/Attraction_Review-g274914-d276838-Reviews-Kovacs_Margit_Muzeum-Szentendre_Pest_County_Central_Hungary.html