I'm planning on doing this route as well in early June. I emailed UDOT asking about the specific location of the closures. Here's the prompt response I received:
Center Street in Cedar City is milepost 0. Gates are in place at approximately milepost 7.2 and milepost 10.So it would seem that the closure it just for three miles. I made it clear i was planning on bicycling, and they didn't say anything about it. FWIW, it appears from their new site (http://www.udot.utah.gov/restore14/) that the road will open on May 24th for Memorial Day weekend until 7am on May 28th. After that, it's closed 7am-7pm on weekdays, and open 5pm on Fridays until 7am on Mondays.
It seems to me if we ride to the closure by 7pm, we ought to be able to make it the 6 miles to Cedar Canyon Campground before dark. You might be able to do those 10 miles in the morning, but that's some pretty steep climbing. The temporary road is unpaved, so perhaps in bad weather you may want to try and hitch. If I have any issues I'll post here - i should be heading through around June 5th or 6th.