What is the sign of mozzarella being old?
It's vacuum sealed, and the cheese looks fresh. Is a month a big deal?
So long as it's been kept properly (refrigerated), it should be safe to eat. But it will have a different flavor than you're likely expecting. Not much, for just a month. But cheese flavors evolve as the cheese ages. "best by" dates on cheeses are mostly about flavor changes, not spoilage.
You can abuse this to a limited extent, if you'd like. You can age cheese at home and (for example) create sharp cheddar from mild cheddar. However, the flavors evolve better in larger pieces of cheese; an 8oz piece aged for 90 days won't taste the same as a 40lb block aged for 90 days. Many cheeses also require specific temperatures for the cultures to age properly, which home environments can't control as well.
If there is mold growing on the cheese, or other "ick" factors, you likely want to toss it. Tiny bits of cheese mold can be removed from a piece and leave good cheese behind; but heavy mold growth on a small piece isn't going to be salvageable.
Mozzarella costs what, $4 tops? Weigh up that cost versus food poisoning and you have your answer. As always, if in doubt, throw it out.