--- Take the 200 bus (its the new airport bus) to FELSŐCSATÁRI ÚT
then change to :
--- 95 Bus and go to final station : STADIONOK (IFJÚSÁG ÚTJA)
--- M2 Metro just one station to Keleti Pályaudvar
This is the route i took end of may. I arrived in the morning. Both busses usually offer enough space even for big luggage (22 KG trekking bagpack). Enter the 95 Bus at the rear there is a lot of room ! Not so in the front. Metro was a bit funny but it was just one station so no big deal. You could also walk maybe.
Check here for alternative routes and different times :
if you need i can post translation ...
Make sure you buy the needed tickets from the first bus driver at the airport or from the machines or news stand because later it will be harder without hungarian.
Of course for most people i think its more comfortable to just get a taxi but if you are on budget or like to be independent its a good option.