Hotel convenient to airport

There is no need to stay near the airport. The check in counter won't be available for your flight until 2 hours before departure. I would select a hotel in the city center and plan to enjoy one last breakfast in Budapest before heading for home.

You have several options to get back to BUD:
1. Taxi - pre-order from City Taxi or Fő Taxi - both companies are reliable and reasonably priced.
2. Use Silverwings: http://www.silverwings.hu/ (as recommended by Mr. É.). When it's more than the 2 of us, this is what we choose. It's a bit more expensive than a taxi, but the drivers are wonderful and it's great for a group with a lot of luggage.
3. Take the 100E Bus - cheap, but it can be crowded.

Christy is right - it saves at most 30 minutes to stay out by the airport. And it’s not pretty or convenient to anything. So obviously my choice is always to stay in the city and just get up 30 minutes earlier.

With 4, I definitely recommend City Taxi. You can set up the app before you leave, and just schedule a pick up the night before. Or have your hotel call the night before. I haven’t used the transfer Christy mentioned since I am always solo, but that would also be a very reliable solution as well. I took the bus last time and it is easy and cheaper, if that matters. Staying near Deac Ferenc ter your last night would help with that.

Enjoy!!

From Pest it's a 35 minute taxi trip to the airport. For your last night get a room on the river with a view.