There is a taxi pickup area by the Terminal Arrival area at Taiwan Taoyuan International Airport. The cost is about NTD1,100~1,700 for a one-way trip to any Taipei destination.
By Public Transportation
From the airport arrival area, take the Airport MRT (Mass Rapid Transit/ Subway line) or bus to Taipei MRT Zhongxiao Fuxing Station. Then, take Taipei MRT brown line to the MRT Taipei Zoo Station. Later, take the bus (#236, 237, 611, 282, Brown-11, or Brown-6) across the street from MRT Taipei Zoo Station to NCCU. The total cost is about NTD 250.