I would like to go to shishanzhen, Nan'an. Can I take direct bus from Xiamen Gaoqi Airport?

There is no bus to Shishanzhen at the airport. You need to take shuttle bus North Railway Station line at T3 Station to Gaoqi Bus Station at first, which usually takes around 25 minutes. Then take a long distance bus to Shishan Bus Station directly.

I would like to recommend you Jinjin Business Hotel near the bus station. It is located at No.65 Tanmei Street, Shaner Village.
May it be helpful.