I would spend the extra money and get some Gunkote or Duracoat. It will last a lot longer and look better. Beisdes if you get any gun cleaning solvents on the barrel the spray paint will come off.
Brownells also sells Gunkote in a aerosol can that works well as long as it is prepped good. You can sand blast it or roughen it with sandpaper then use some non clorinated brake cleaner to get rid of grease. Then heat the part before applying the Gunkote per the instructions.
The way I clean before Gunkoting is to bring some water to a boil, remove it from the heat and add simple green. Put the parts in for about 10 minutes, sloshing them around every few minutes. I remove the parts and pour out the simple green, rinsing the pot well. I again boil water this time I put the parts in and sloch them around to remove all of the simple green. I do this step twice to make sure all the simple green is removed from the parts. When I remove the aprts from the boiling water they will dry themselves in a matter of minutes becuae of the heat they have absorbed from the boiling water. When the parts are completely dry I spray them with Gunkote per the instructions. After spraying I put them in the oven per the directions.
Because it is a barrel you may have to use the brake cleaner rather than simple green.
I have never had any problems doing it this way. WIth Gunkote follow the directions precisely or you will have problems.
Dolomite