Forget the sleeve, rebarrel it. As long as the action is in good shape. Decent barrels can be had for $100 or so and then pay a gunsmith to chamber, finish, and install it. (another ~$100) You have several cartridge options besides just the standard 8x57js. 7x57, .257 roberts, and 6mm remington all share the same parent case. .308 is a popular barrel swap as well but any cartridge that uses the same bolt face diameter and isn't too long for the magazine will work.
Then what I personally would do is buy an aftermarket trigger/safety, good looking walnut sporter stock for it and have a nice "custom-ish" hunting rifle. But thats just me. Some prefer to keep it original but I love sporterized Mausers.
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