There is an airport shuttle that will drop you off anywhere you want in Budapest. Compared to the subway system, in terms of convenience, the airport shuttle is EXCELLENT value. I would highly recommend it.
Here is a link -> http://www.airportshuttle.hu/en/ about 5 years ago
Besides the shuttle there is also public transport, first you need to take bus #200 to the nearest subway station (KŐBÁNYA-KISPEST), bus runs like every 10 minutes. While from Kőbánya-Kispest you can either use subway or other bus to almost any part of Budapest about 5 years ago