Typically, I just refill my water bottle at a public fountain, but otherwise, I would pay $30NTD ($1 USD) for a plastic water bottle.
My favorite meal was at a famous soup dumpling restaurant called Din Tai Fung. They had a wide array of dumplings and entrees that were vegetarian-friendly, and the restaurant had amazing service and complimentary tea. It used to be a Michelin-starred restaurant!
Taipei is a very international city, so there are places to listen to K-Pop, Latin, Taiwanese and American songs.
We had a college club fair that showed off all kinds of different student organizations and small businesses that advertised themselves. NTU has a very international scene, as there were even jazz and tap-dancing clubs.
I haven't done much reading in my free time lately, but a good book recommendation is Speed of Trust, by Stephen Covey. This is a great and practical book for building up interpersonal skills.
NTU has an outdoor calisthenics area with lots of bars and poles. Calisthenics are a great exercise for people looking to build a lot of muscle mass, without having to worry about too many people in the gym. They're all bodyweight exercises that especially help your mobility.