To clarify here are the possible routes I can think of, excluding buses.
Caveat is that trains don't run between 12am-6am so if you arrive during that time, taxi is the best option.

1. Taxi. About 1140-1180NTD according to 55688 taxi app. 35-50mins depending on traffic, but since you're not crossing into Taipei proper I don't think this should be too bad.
2. Airport MRT https://www.taoyuan-airport.com/airport_mrt?lang=en to Taoyuan HSR Station (A18) for 25 NTD. Take HSR from Taoyuan to Banqiao for 130 NTD. Total 155 NTD, about 45 minutes. This depends on you making the transfer efficiently to the HSR, otherwise the wait between trains might make this much longer.
3. Airport MRT to New Taipei Industrial Park station for 105 NTD. Take MTR yellow line to Banqiao for something cheap, let's say 25 NTD. Total 130 NTD and about an hour. This depends on you getting the express Airport MTR or it takes longer.
4. Airport MRT to Taipei Main Station for 150 NTD. Take MTR blue line to Banqiao for 25 NTD or take Taiwan Railroad for maybe a little bit more. Total 175 NTD and about an hour fifteen. I think this is longer than Option 3, and the transfer in Taipei Main Station is a lot of walking. Post 2 is referring to this.

It depends on your mood. If you have a lot of luggage, taxi might be best. Option 3 is more regular than Option 2 but if you're a gambler or like high speed trains you might want to go with Option 2. Option 4 seems less efficient.