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Sleeves can work, but they require an extra halyard (which you probably have) Another factor with sleeves is that in breezy conditions they can start to flap around and can chafe the exposed stitching of the rolled up sail. Some put elastics every 4 feet or so to discourage that, but then if its windy enough you'll get advance, localized chafe in way of the elastics.
Your best bet is probably to remove the tattered bits and have a new cover installed... or, you can do as we do (furler with no UV cover on the sail) remove the sail and store it below between trips, just as you would with hanked on sails.
Though it's a little more work, but costs nothing and there's no weathering on the sail at the dock.
1984 Fast/Nicholson 345 "FastForward"
".. there is much you could do at sea with common sense.. and very little you could do without it.."
Capt G E Ericson (from "The Cruel Sea" by Nicholas Monsarrat)