I recently stayed at the Bunkhouse. Great included Irish breakfast in the morning, free wifi, clean rooms and good bathrooms, full kitchen facilities.
The 41 bus starts earlier, at 5 AM, from Lower Abbey Street. There is a bus stop on Upper Gardiner St not a far walk from the Bunkhouse (where I stayed). From that stop, it's 1,80e to the airport. It will actually get you to the airport 10 minutes earlier than the first Airlink bus. However, if your flight is at 6AM exactly, that is still too late. You'd have to get to the airport the night before, or pay to take the AirCoach for 7,00e.
The good thing about taking the Airlink FROM Dublin Airport to the city centre if your flight arrives early in the morning is you are actually purchasing a Day Rambler that will then allow you unlimited travel on Dublin Bus busses for the rest of the day.