Possible. You will need to apply for a ARC after you enter Taiwan.
Information and the required form can be found here: National Immigration Agency, R.O.C. (Taiwan)-Notices for foreign nationals applying for an ID No.
Office in Charge: Local Tainan Office of the National Immigration Agency
Tainan City First Service Center:
1. Agency Address: No.262, Section 1, Fu-Chien Road, West Central District, Tainan City, Taiwan (R.O.C.) (台南市中西區府前路一段262號)
3. Map: 臺南市第一服務站
With that number, you may open a bank account in Taiwan. The Campus Post Office is recommended due to its convenient campus location.