Is there a bus that goes to/from the airport to the Buda side of Budapest? Also, how would one go about traveling to Banska Bistrica (Beszterszebanya) in the Csech Republic. As I recall, it is due north of Budapest, just over the border.
http://www.bud.hu/english/transport/airport_shuttle
Note: The airport website also talks about the other possible routes....Enjoy Budapest
I can offer you a partial response based upon my own research for my trip for Budapest. I cannot say for sure if there is not a specific bus that does what you want
The four principle means of transport from the Airport to Budapest
Taxi - Zona Taxi is reccomended.
Shuttle Bus - Group buses that will take you to any address in Buda / Pest. Note: If you are in a party of two you will pay the same as for a taxi anyways but may not go direct.
There is a Bus (200E I think) that runs from The Airport to Kishpest - South Terminus of the metro system. You can then take the Metro in.
Train runs from Airport to an Urban Station on the Pest side. you may be able to find a bus from there to where you want to go.
Do you know the name of the shuttle bus company?