The Buda Castle can be reached on foot from the city, by cable car, bicycle, but also by city bus that crosses the bridge on the right side of the castle. For only 1.00 eur. On the other side of the river, small blue carts with the inscription "Official Budapest Castle bus" will take you to the castle. This bus costs 9 euros (return ticket) or 2700 forints and departs every 15 minutes. Budapest Card holders have a free ride.
