Or, this may suit depending on where you are staying:
200E bus to Kőbánya–Kispest metro terminus, then change to the metro and take the M3 line towards Újpest-Központ. (This route suits me since I'll be staying along the M3 line between the stadium and the city centre.)
BKK ticket machines are just outside the airport door, as are the 100E/200E bus stops.
If anyone's staying for a few days and needs to hop on and off transport, then the "Budapest 7-day Travel card" may be worth getting, and it costs HUF4950 which is about £14 and includes the airport transfers plus all trams/buses/metro.
http://www.bkk.hu/en/tickets-and-passes/prices/
http://www.urbanrail.net/eu/hu/budapest/budapest.htm
I gave up waiting for us to draw Hungary and took a city break last autumn