What is the shelf life of fresh baby mozzarella in water ... once the container is open?

Treat it the same way you would fresh fish. Just don't freeze it.

I usually just change the water everyday-- Put a bit of salt in the water and keep your mozz covered. It will probably stay fine for several days.

it usually says on the packet up to 2 days, and I'd go with that because after that it starts to go hard as the air gets to it