Yes. When you renew your passport, you need to update your APRC to reflect your new passport number. Make sure that you take two new pictures, your old passport and your new passport with you when you go to update your APRC. I also believe it will cost you $100 ntd in fees, if I remember correctly.
Also, you should have an Open-Work Permit along with your APRC. You must also update your Open-Work Permit to reflect your new passport number as well. See the
Open-Work Permit Application Procedures - Step by step Permit thread for specific instructions. 
I had to renew my passport last year and had to update both my APRC and my Open-Work Permit. Major time consuming pain in the ass, but unfortunately necessary.