Bruce Gordon Ti Rock N Road , Fat Chance Mountain Tandem 
Long Canyon and the Burr Trail are a great choice. Just beautiful. I would call the ranger at Boulder,Utah or at the Capitol Reef Headquarters for current conditions.
The road surface changes with the degree of rain each year-rockfall, exposed gravel and imbedded rocks. If you are heading East the distance between the pavement end and next paved portion is under 20 miles as I recall. [Maybe more of it is paved by now! ]
The main problem that I had there a few years ago was at the East end of the Burr Trail- deep sand through the E-W notch crossing the Water Pocket Fold heading toward Bull Frog Basin. I could have used a wider tire with more air volume for more float in the sandy portions.
Here's a pic from a few years back - October not May.
May should be fine, but the high country can get a late snow.
And that dirt stretch - - includes a big set of hairpins downhill if going east -
Steep uphill if heading west.