You are an exchange student, so don't worry about tuition and fees. I believe that living in normal university will also arrange to live in the dormitory of the school. It will be very expensive if you don't arrange it, because in Taipei, a 1 month room is about NT$ 6,000-8,000. Next to Normal University is the famous Normal University Night Market. You can eat there. Generally, a box lunch costs about NT$ 50 in 65 yuan and NT$ 35 in a local breakfast chain in Taiwan Province Province. Eating one meal a day won't exceed 200 yuan NT. Only NT$ 6,000 a month. About RMB 1300.
Transportation can be done in easycard. If there is no special need (such as going out to play often), it is 200 RMB a month.
Part-time job is more difficult. Generally speaking, the purpose of your visit to Taiwan Province Province is to study. If you are caught working outside the home, you will probably be sent back to the mainland (which is not in line with your purpose of coming to Taiwan), and it will affect the impression of Taiwan Province visa officers on you next time you come to Taiwan. But you can be a tutor. Many students in Normal University are tutors, maybe teaching math or English. They can earn 500-700 Taiwan dollars per hour, which is a good choice. Just keep a low profile and don't be reported by your parents.