From Taoyuan Airport to Green World Hotel by MRT and HSR

Take the Airport MRT (stop at both airport terminals), to go to Taoyuan high speed rail station.
(it took ~20 min)

Airport MRT (Taoyuan MRT) map  Another Airport MRT map  (A12 (terminal 1) or A13 (terminal 12) to A18 (Taoyuan HSR station))

At "Taoyuan HSR station" of Airport MRT, switch to high speed rail (~ 5 min walking distance).

Then, take high speed rail to Nangang station, which is also the last (or first) high speed rail station.
(~ 30 min, 3 stops), you should buy ticket at ticket counter before aboard (buy Reserved Seats ticket if it is busy hour, non-Reserved Seats car is always full with people at busy hour).

Then take Taipei MRT Bannan (Blue) line from Nangang station to Houshanpi station (two stop, ~ 3 min) GreenWorld hotel is 40 meter from exit 3 at Houshanpi station.

Info on Taipei MRT  (Take BL22 (Nangang) to BL20 (Houshanpi))