Does alcohol-free beer go off / have a shorter shelf life?

Alcohol is a natural preservative, so removing most of it can mean alcohol-free beer has a somewhat shorter shelf life than its full-strength counterpart - but "shorter" does not mean short. Most alcohol-free beers are still good for several months to a year unopened, and the can or bottle will always carry a specific best-before date, so there is no need to guess.

Store it the same way you would regular beer: somewhere cool and out of direct sunlight, and drink it reasonably promptly once opened, since the lack of alcohol means it will not resist going flat or stale as well as a full-strength beer might. If in doubt, check the date on the can rather than relying on general rules of thumb - shelf life varies by brand and production method.