Parliament in the morning, the Shoes on the Danube Bank, St Stephen’s Basilica and its dome, then the Advent market at dusk.
Day 2
Buda Castle
Funicular up, Matthias Church and Fisherman’s Bastion, the Hungarian National Gallery, then down to Gül Baba’s tomb.
Day 3
Andrássy & City Park
Walk Andrássy út past the Opera to Heroes’ Square, the Museum of Ethnography, then an afternoon at the Széchenyi baths.
Day 4
Market & Jewish Quarter
The Great Market Hall, the Dohány Street Synagogue and the Jewish Quarter’s courtyards, then Rudas baths at night.
Day 5+
Beyond
Szentendre or Esztergom for the Danube Bend, Gödöllő’s royal palace, or the train to Vienna.
Honest thought: daylight ends around 16:00 in December. Do outdoor sights in the morning, museums after lunch and save markets, baths and river views for the dark, when the city is lit.