How can I sharpen a round leather punch?
The approach is the same. You remove metal from the outside of the periphery of the punch end, at an angle roughly the same as it was originally.

The contour of the round punch adjacent to the cutting edge looks rounded in your picture. This makes the edge stronger, but makes it trickier to get the sharpening angle right when using a flat stone. The punch won't get sharper unless you remove material right up to the opening, producing a keen edge.

You'll have to pick your sharpening angle (I'd start with 45-degrees) and stick with it until your grinding marks meet the opening. With something like this it's sensible to mount the punch in a drill or drill press and use the stone against the edge while it's rotating. Best results come from holding a consistent angle, which is easier when the drill or other rotating device is fixed to a bench or whatever, not hand-held. I suppose you could try holding the stone fixed and run a hand drill guided by hand, but avoid holding both in hand.

If you're brave and have a steady hand, you could try using the hand-held drill approach against a a bench grinder or belt sander instead of a stone. Only problem there is when things go wrong the go wrong quickly. Might be something to work up to once you've gotten a feel of how it works against a stone.