How do I go to Taipei city from Taoyuan international airport?

1. From the airport, you can take direct coach to Taipei. The ticket fare is NT110 to 140 per person. It takes 1 to 1.5 hours on the way. Or you can take shuttle bus to get to the Taoyuan Railway Station first. Then take high-speed train to Taipei. The duration is about 20 mins. The ticket fare is NT 175 per person.
You can choose way you like.
2. As I know, there is no such tourist pass for the island tour.