It is kind of an awkward location for the Metro - dragging your suitcases in Taipei summer weather a few blocks can be challenging. The Xing-Tian Temple Taipei MRT stop (O09) seems to be the closest but it is still a few blocks away (my guess a 15 minute+ walk with suitcases and heat).
Taipei Metro Map:
https://english.metro.taipei/cp.aspx?n=1BE0AF76C79F9A38 and Exit #2 is the best exit to use at O09:
https://web.metro.taipei/img/ALL/Route2200/131.jpgThe hotel is just past the elevated Jian Guo road.
It seems the most direct route on Metro is to take the Taipei Metro Orange Line to San Chong (O15 - note that the Orange Line splits so you want to take the one that goes to O15) then get the Airport Taoyuan Metro. Note that the TY Metro express train does not stop at San Chong but you are not losing a lot of time.
Evergreen also has a bus service that goes to Min Sheng E Road and Song Jiang Road (bus 5201). Some buses are Hello Kitty
Evergreen Bus Map (hotel is between stops 2 and 3) &
E-Bus Page for 5201 (Kind of Slow)