driving from budapest to prague airport

It's about 350 miles and will take somewhere between 6 and 7 hours depending on the route you take, delays, traffic, etc.

I wouldn't do it. But if you insist, don't leave any later than 5 AM.

If you rent a car in one country and drop it off in another, you’ll have to pay a very large drop-off fee.

on top of the valid warning from lnbsig there is also the matter of insurance, you need to make sure that a Hungarian car has insurance for Czech Republic AND either Slovakia or Austria depending on which route you take. And dropping a Hungarian rental at Prague may take longer than simply handing over they keys. Obviously this all assumes that you are using a rental car - and that the company will allow it out of the country even if you do offer a pile of money on top of the usual rate.

And dropping a Hungarian rental at Prague may take longer than simply handing over they keys.

I agree. If anything happens to your car either before or during the drive, there will be extra time needed to go over the damage, fill out extra paperwork and so on. That happened to me in England. The agency was nice enough to drive me to the train station afterwards where I needed to catch my train, but I barely made it. I was literally running once I got to the station.

Like the others said, I wouldn't do it. It would be so much stress if you get delayed. It's not worth it. My suggestion would be to drop off your car in Budapest (if you rented it in Hungary) and take the train to Prague. But if you rented it in the Czech Republic, then drive to Prague the day before and spend the night there. No stress and you can have a relaxing evening in the city, then leisurely make you way to the airport the next day.

Train Budapest-Prague would also require travel previous day (or a dreadful night itinery). Flixbus have an overnight direct centre to centre bus which is a slightly better itinery and cheap enough, but you're only on it for 6 hours so how much sleep you would get is a question to be answered afterwards. There is an express bus to the airport from the main railway station in Prague, which is probably why Flixbus call there as well as at the bus station.

Flights are faster than the train. Sunday, Wednesday, Thursday and Friday there are flights that would get you to the airport early enough before your flight at 2pm, but if anything went wrong you would miss your flight at 2pm. I would fly to Prague the day before. Or Daytrip.com will cost you just under $500 for the trip.