The Bus Terminal is not open for 24 hours. Furthermore, there is no waiting- or restroom inside the bus terminal. So you better stay somewhere else. Have you considered staying at a cheap Hostel, close to Hakata Station than take the first Subway the next morning to the airport?

The earliest bus departs 6:30 am from Hakata Bus Terminal Platform No 11. You will arrive at the international terminal around 6:45 am.

Nishitetsu runs bus from Hakata bus terminal to Dazaifu station. Nishitetsu runs train from Tenjin more frequently, which I usually ride and is faster than the bus.