There is water at the visitor's center (well, there is at the campground, which is near the visitor's center). There is also water at Devil's Kitchen picnic area on the other side of the park.
They are now requiring lights in the tunnels. Check the "bicycle regulations" on the website if you haven't already: http://www.nps.gov/colm/planyourvisit/bicycling.htm
Have fun. I LOVE the monument!