So you have 5 days.
Let’s say you take the flight that arrives about 11pm in Budapest. Get to your digs, get comfortable and go to sleep. Or wander a little if you like. For that you will want to be staying along Andrassy ut or Kiraly ut or down in the Fifth District if you like being in the center of the tourist drag. All great places to wile the evening away. No matter what I will assume you are staying in Pest somewhere near or close to Deak Ferenc ter.
MORNING: Buda Hill
AFTERNOON: Nyugati Station, Oktogon, upper Andrassy, House of Terror and Hero’s Square.
MORNING: Parliament, Falk Miska utca, No 2 tram along the embankment.
AFTERNOON: Market Hall, Vaci utca, Basilica
Day Trip to Szentendre. Train there and the boat back
MORNING: Kiraly utca, Great Synagogue tour
AFTERNOON: Jewish District and the lower end of Andrassy ut
MORNING: Museum Day: Trains, Transportation, Art, History, Military. What’s your interest
AFTERNOON: Bath House
EDIT: Each of the places listed above is as much about the neighborhood as the place named. For instance, "Bath House" is City Park, the Zoo, the lake, possibly boating on the lake, etc.......
Wine Tours (one day or overnight)
Overnight to Pecs (amazing town. Google it)
Day trip to the Archabbey at Pannonhalma and the town of Gyor
Overnight to Eger the wine town and the Tokaji wine region
Drive a Russian main battle tank
Delve deeper into the Jewish History of district VII and VIII
Day trip on a vintage train (coal burners and early diesel)
This could go on for 10 years……….