To reach Pomàz and the Choral Castle you will find all sufficient information at the following website:
http://www.choralcastle.hu/index.php?option=com_content&view=article&id=141&Itemid=114
When arriving by plane the most adviseable solution is to take the shuttlebus called "Airport Minibus", it takes you directly to the Castle. More info at http://www.bud.hu/english/transport/airport_shuttle