Youbike’s relatively new - and its range is certainly new - but I’m not sure how much that’s worth checking out for the OP. Are there any good YouBike routes through across the city? I can see potential for routes across town, but that potential would rely on avoiding Taipei traffic, and I’m not sure that’s an option.
The riverside cycling paths have expanded somewhat since ten years ago, although even ten years ago I think most of them were in place. Those are definitely nice for a few hours, weather permitting. Riding YouBikes one-way between Guandu and Danshui may be the best short trip on that; from Danshui around to Bali is a nice slightly longer route. YouBike has a fairly good iOS app for finding stations and bike availability.