As private cars are not allowed to enter the zone, your only options remain public transport, walking or taxi.
Szell Kalman ter, Buda's main transport hub station M2, makes the most convenient entry point to the Castle District. You can walk upon the hill it takes about 20 minutes or take the bus 16 or 16A. When you come up from the subway station look for the outdoor stairway leading to a pedestrian bridge over the trolley tracks. Up the steps, over the tracks, cross the street to the mid-block bus stop.
It's also pleasant to walk by Chain Bridge Széchenyi LánchÃd from Pest and walk up by the narrow streets or paths as Király lépcsÅ or Kemál Atatürk sétány. Walking from Pest is likely to be faster than public transport or taxi on peak hours.
The spectacular but very touristy and pricey cable car funicular called Budavári Sikló HUF 900/1500 one way/return, children HUF 550/1000 climbs up Castle Hill from Chain Bridge's Buda side to Fisherman's Bastion Halászbástya. As the cable car is rising up the hill you get a beautiful view of Pest across the river. Alternatively you can check the view from one of the two small pedestrian bridges over the cable car way - for free.
