I would say, it's not very likely. There are four distinct parts that a revolver cartridge consists of - a bullet, gunpowder, case and primer.
Bullet is easy enough - lead bullets were used in the slings since the Roman times, so casting lead to acceptably uniform size and shape is not that hard.
Gunpowder is harder. At that time (10th-12th century) gunpowder is not familiar in Europe yet, it will come only in the middle of the 13th century. I think your gunsmith would be able to manufacture the gunpowder for himself, if he has enough knowledge of chemistry, but that will be a time-consuming and difficult undertaking.
Primers are harder than that again. Mercury fulminate was not known until the 19th century. It's production is also in the scope of a single person with access to resources, but the necessary knowledge of chemistry is greater again, since your gunsmith would need to produce nitric acid, pure mercury and ethanol first under high medieval conditions. The chemical knowledge on its own moves your character out of 'simple gunsmith' to 'gentleman naturalist' category. Building the chemical laboratory and gathering all necessary resources would be difficult too.
Surprisingly, making a case itself seems to be the biggest problem. I have found this description of cartridge manufacturing process from the 19th century. As you can see, you need to have uniform plates of brass of consistent thickness first and then several machines to form them into cartridges. I do not think that making all that by hand without building the machines first is possible - even such early cartridges as .45LC have pretty close tolerances.
In conclusion, I would say that the scenario you describe is pretty unlikely. It would be slightly easier for your character if he wouldn't need to manufacture his own brass - either reloading old cartridges until he runs out, or using an earlier percussion revolver like Colt Dragoon.
But even in the case where the character doesn't need to make his own brass, he still needs to manufacture both black powder and primers. For that he needs (1) to be extremely lucky to be able to establish an alchemy lab, (2) to have enough social connections to be left alone to pursue his alchemy experiments, (3) to have access to necessary chemical resources and finances and (4) several years time to refine his process and get first results.