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The steel yellowish pendulum (I call it big ball) in the photos above serves as a tuned mass damper, the ball sways to offset movements in the building caused by strong wind (if you know that Taiwan is prone to high winds from typhoons as well as earthquakes). It is the largest damper sphere in the world, and it is also the one and only damper sphere which is visible to visitors (tall buildings like KLCC usually have a damper sphere installed to offset movements in the building). Therefore, the Taiwanese are very proud of it :)
Here's how to get to Taipei 101 Observatory by MRT:
1) Get off at City Hall Station (Blue Line).
2) You can either catch a Taxi to take to there which will usually costs less than NT$100 or you can walk from the MRT station to Taipei 101 following the google map attached. It is just 10-15 mins walk from the station.
View Larger Map
Here is the MRT map for your reference:
Click on the map to view the full size image.