from Kaohsiung Airport to Tainan city by taxi how much the cost?

Hi, 300 NTD can be ok.

300nt? Not even close. Sara doesn't have a clue. From Kaohsiung Airport to Tainan City it would be at least 1000nt. Don't even think they would take you that far, since it's another city and district -- so you would either have to pay double... or transfer over to another taxi after.
Best thing to do is just take taxi to train station (HSR - high speed rail - is not in Tainan city).
Takes only about 30min by train.