When my carpal tunnel syndrome act up should i have the affected hand rest completely or should i try to do some exercises with that hand?

First try PT: Physical therapy will help some , try first , if you are not getting benefit from PT other only alternative go for surgery , which is simple low risk out patient procedure

Some advice: We usually find that repetitive flexion and extension activities at the wrist promote carpal tunnel syndrome, and one approach to help, involves usage of a wrist extension brace, which keeps the wrist in a constant position. Exercise really is of little benefit if you have advanced carpal tunnel and injection or surgical decompression is highly successful.