Came back from Taiwan two weeks ago.
Eva Air has a bus division (長榮客運) that offers transportation to downtown Taipei (really close to Songshan) for about NT $150. It runs until 1am.
Here is their list of stops if you can read Chinese.
http://www.evergreen-eitc.com.tw/eit..._4/c_4_3_0.jsp
Bus tickets can be purchased at the main bus counter (many options to choose from). When you exit out of security, go toward the right. You will see a right hall way that leads to the exit. Bus counter is right before the exit.
Or just follow the sign at the airport for ground transportation.