From Terminal 1 (the LCC terminal), you can take a direct local train to Nyugati station in the city center in 23 minutes for 365 HUF.
From Terminal 2 (used by everybody else), your best bet is to take bus 200E to Kőbánya-Kispest and connect to the metro (Line 3). A through ticket is 320F in advance, or 400F from the driver.
Credit:stackoverflow.com‘