You can take the MRT from the airport to the city center, or you can take the bus to the city. The MRT is faster and the price is a bit higher than the bus.
Downtown Taipei? Do you mean Taipei Station or Ximending? If you are taking the MRT at Songshan Airport to Zhongxiao Fuxing Station, transfer to the blue Bannan Line to Taipei Station or Ximending.
Songshan Airport has Songshan Airport Station, which is on the MRT Neihu Line, which is the Brown Line. The Taipei MRT extends in all directions, and you can transfer to any line to go to your destination in downtown Taipei.
If the city center you mentioned refers to the vicinity of 101, then you can just take a taxi, and you can get there for about RMB 50. Songshan Airport is in the city.