One does not really arrive in Taipei on an International Flight but at Taoyuan - CKS Airport. It is about 45 mins away from Taipei City.
The best way plus the most economical way to get to the City is by one of many Airport buses that take you to the Taipei Main Station with drop off points along the route at Major Hotels and MRT Stations. Once in the City, you can cab it to your Hotel if it is not on any one of the designated routes. While language can be a problem, you can call for an English speaking Taxi Driver at no extra cost.
List of English Taxi Drivers and their contact numbers
Toll free hotline for taxis (For English service please press two)