From The Airport
Taoyuan International Airport (CKS Airport)
You can take buses of Kuokong Motor Transport Company, Airbus or any bus that goes to Taipei.  It might take you about 50 to 90 minutes (about 50 km) to Y Hotel Taipei depending on local traffic flow.  Taxies are another transporting options; taxi fare can be negotiated(about NT$1200 as estimated). 

Songshan Airport 
By Bus:You can take bus No. 262 or No. 5 and make Taipei Train Station your destination.  That will charge you NT$15. 

By MRT: you can take Wenhu line from Songshan Airport Station to Zhongxiao Fuxing Station, then transfer  Bannan line to Taipei main Station. 
By Taxi: it costs around NT$250

From Taipei Main Station:
TRA,MRT,HSR  stop at Taipei Main Station , Go south-east direction, you shall find MRT Exit 8 ( M8 ), then find XuChang St. on your left. YMCA is a 3-4 minute's walk from M8.