Buda used to be an important point for Turks - further territories could be conquered from here. They destroyed much more than built, turning churches to mosques. However, you'll discover wonderful bathhouses here with arch domes. Visit Király and Rudas to be sure that Turks enjoyed themselves in Buda.
It is also worth climbing Gül Baba utca to see a nice garden with a fascinating view of the city. Turks turned the place into a mausoleum of dervish Gül Baba's (Father of Roses in English), who arrived in Buda in 1541 to be in charge of local religious life but died almost right away. There are two versions of this. The famous dervish either died during a prayer of thanks to the Turks' victory or was killed at the siege of Buda. Any way, his funeral was flamboyant. A small garden round the mausoleum proves the point. There is no city fuss here. You may just relax in the shadow of flowering trees, drink from the murmuring fountain and see the city from the above.
After catching your breath, continue ascending the hill. This is a fashionable district Rózsadomb with villas and small apartment houses with gardens. There are no tourists here, while locals are considered to be very wealthy people.