I'm going to Taiwan; Taipei, Hualien and Nantou from 26 Apr 2016 to 1 May 2016.

I just checked the latest weather reports. Light rain will fall upon the three places all the time during that period, except that Taipei has a sunny day on Apr. 29. To my experience, the rain won't last for a whole day, so cycling can still be on your list, but it is best to avoid mountain areas for the sake of safety.