A great way to see Taipei and the green hills and mountains bordering the city while getting your caffeine fix is from the Starbucks outlet located on the 35th floor of Taipei 101.
Prior booking is required- at least a day ahead. You can either call +886 2 8101 0701 or request hotels or tourist information centers located at major tourist attractions and some train stations to call and make the arrangements for you. I suggest going through the information centers as communication might be an issue.
My friends and I made a booking at the information counter of the Taipei 101 shopping mall which called and made the necessary arrangements.
Arrive early and wait at the sign which also outlines the requirements to make the booking. Was definitely worth our time. Holding those hot cups of coffee, tea and red velvet beverage on a cold winter morning while taking in the view was indeed a pleasant and memorable.