Finally, the authorities have solved the problem for some travelers who find it difficult to get and from the airport with public transport:
MÁV Hungarian railways has launched a new rail service connecting Terminal 1 of Budapest Ferihegy airport with Nyugati station.
The 25-minute journey is charged at the normal, per-kilometer rate. A single ticket costs Ft 300 (€1.20) and trains run twice or three times an hour. MÁV is also planning to schedule InterCity trains traveling between Budapest and Cegléd via the airport.
Just to emphasize, this is only for Terminal 1, where the budget airlines operate out of, not 2A (Malev) or 2B (int'l carriers).
This just started, so I'm not sure where one purchases the tickets at the airport.