How to get from TPE airport to Diary of Taipei hotel in Ximending. How long is the journey?

You can take the airport shuttle bus to reach Taipei Railway Station. Then you can change to the local metro to reach Ximen Station. Get out of the station from Exit 6.
Come out of the station, turn right towards the main road (zhonghua rd), walk around the building and you'll see a small banner that says "diary of taipei hotel" in front of a small elevator lobby. Get into the elevator to reach the 5th floor.
Or from the railway station, y ou can take a taxi to reach your hotel directly, because it is not very far.