Walk the compact medieval Upper Town to see St. Mark's tiled roof and the Parliament building; great orientation walk and photo spots. St. Mark's area accessible ~9:00am–5:00pm.
See the exterior of the Hungarian Parliament (book tours in advance if wanted) and the poignant 'Shoes on the Danube' memorial; walkable and mostly free.
If wanting to save, take a 1-hour local train to Bratislava for a low-cost cross-border day-trip; otherwise visit Prater or Hundertwasserhaus in Vienna.