How is it best to get to Fuzhou airport from Putian, we need to arrive 05:00am for flight at 8am?

You may take a direct bus to the airport from the local Tianjiuwan Passenger Transport Station. The first bus is scheduled at 5:00, and the trip takes about 2 hours. The fare is CNY 52. Generally speaking, you need to reach the airport at least two hours before departure. So, you may consider taking a taxi, or staying near the airport. :)