Okay, so my flight home generally leaves at about the same time and I do it two to three times a year; and have been doing the same for nearly 15 years.
The night before I call City Taxi +36 1 211 1111. Why City Taxi, cause its a known and they have never let me down. They have actually done some pretty amazing things for me over the years so they get my recommendation and my business. It will cost you about 8.000 forints and the traffic is pretty light at that time of the morning. When you call, first you will hear a recording in Hungarian, then it will repeat in English. Then a human will come on the line in Hungarian. Simply say "Good Evening" or what ever the time of day dictates, and they will switch to perfect English.
For a 6 am flight i would schedule the taxi to pick me up at 3:45. The taxi always appears 5 to 10 minutes early. In the worst instance leaving at this time will get you to the airport at 4:30. The counter doesnt open before 4:30 for a 6 am flight so getting there earlier doesnt really pay off. They wont even tell you which counter to go to until at least 4:30 so you cant even que up. Well I can, cause I know which counters are used by which arilines from having done it so many times. But you do want to get there by 4:30 cause the counter area can be sort of hectic.
No way I would do anything less than 2.5 hours for a change at Chucky D or Heathrow. Amsterdam, Frankfurt and Munich; 1.5 is "okay", 2 hours provides me a lot more comfort.