The Buda Castle and its surroundings, the Castle district are considered as attractions in Budapest that you shouldn't miss. It's part of the UNESCO World Heritage since 1987. The Castle, built in the 18th century is inhabited by the beautiful paintings collection of the National Gallery.
You should check out the famous neo-Romanesque Matthias Church, and the Fishermen’s Bastille, from which – as they say – you can enjoy the most spectacular view of the city. Have a walk in the historical residential area and enjoy the view of the surrounding hills.