The whole reason to come to Bijapur is the architecture of the Adil Shahi sultanate, scattered across the old city. A full day will let you cover the major monuments at an unhurried pace. The big three — Gol Gumbaz, Ibrahim Rauza, and the Jama Masjid — are essential. After those, the unfinished tomb of Bara Kaman and the giant Persian cannon Malik-E-Maidan are the standout secondary sights.
Most monuments charge a small fee (some are free, some 25 rupees, the Gol Gumbaz around 300 rupees for foreigners). The Archaeological Survey of India maintains them; their attendants generally speak good English and the captions are clear.