Until know you have only been able to buy a very limited amount of public transportation tickets at Budapest Airport, but that is about to change.
Last week I noticed a temporary stand that has been placed in Terminal 2A where visitors can buy quite a lot of different public transportation tickets such as 1-day card, 3-day card, 7-day travel card, some group cards and single tickets. This is a great improvement, because earlier you had to buy single ticket at the airport and travel to KÃ¶hanya Kispest, and only there you could buy a one day pass, three day pass or a pass for one week.
At the moment this is a temporary stand, but as I asked them they said that in a while they will get a permanent stand looking nicer with an even larger amount of tickets for sale. So, if you want to travel from Budapest Airport using public transportation instead of a private airport transfer, then this makes it a whole lot easier.