These guys do some excellent offbeat tours: https://beyondbudapest.hu/Tours The cemetery tour is actually creepy and interesting at a lot of levels.
As for Prague, its a bit far if you only have 4 days. You say 4 days: that could mean arriving about noon on one day, two days site seeing and leaving on the fourth day. That's a pretty bare minimum to see Budapest.
Szentendre is an option for a day trip, but at that time of year I might be more inclined to go to the main square in Obuda. It will be decorated for Christmas and have more open. A lot of the charm of Szentendre for me is the boat ride back, but it doesn't run in December.
IF you do have more time and you want something not creepy, but edgy and off the beaten path, you might fly out to Kyiv. Cheaper fight than Prague ($60 vs $120), hotels half the cost of Prague, less touristy than Prague but incredibly fascinating; including ancient catacombs to tour if that helps. For the average interests its a good two night trip. If the subjects interest you it can be a 3, 4 or 5 night trip.