I'm in Taichung and need to get a very early flight from Taipei airport.

According to your departure time, there is no bus that can take you to Taoyuan Airport.
You can take high-speed train from Taichung High Speed Railway Station, get off at Taoyuan Railway Station. The last schedule is at 22:55. It takes around 40 mins on the way. After arriving, take a taxi to go to the airport directly.

Thanks for the information. I've also been told by a local there are buses that run hourly all night and he printed out a couple of timetables for me, so i'm hoping it's correct!