Tracy--I just arrived home today from Budapest, had a 6:30 am flight and I just call City Taxi, this is my 3rd trip to Budapest and they always provide quick service. I stayed near the Liberty Bridge and called at 3:55, they arrived at 4 and I was at the airport by 4:30--you can also schedule a time. +36 1 211 1111. They have operators who speak English, and the cabs accept cash or credit card.
I have to comment on the "leisurely breakfast": there are several places to get lattes, muffins, etc and I certainly chose the wrong one (it's at the back end, with a self-serve pizza place next to it) as it was staffed by the most phlegmatic woman I have ever seen, just her and no one to help make the coffees though there were about 25 people in line. Sort of like going to the post office and they just work at a snail's pace, no sense of urgency or efficiency at all. I had just enough time to drink my latte as I scurried to board, had to eat my muffin later.