How do I identify the breaker a non-working outlet is on?
Short answer:  You don't really know why that receptacle doesn't work, and the simplest thing is to just shut off the main.
If it's a bad receptacle as you are assuming, you can't test it reliably outside the box.  That puts you in a bit of a catch 22;  you can't figure out which branch circuit breaker the receptacle is on, so you can't shut off the branch circuit breaker so it's safe to go inside the box.
If the problem is with the neutral connection, a non-contact voltage in the ungrounded slot in the receptacle will indicate voltage.  If that's the case, you could flip breakers until that goes out to determine which breaker that receptacle is on.  It's not wise to put too much confidence in a non-contact tester, so you'd want to proceed with great caution opening that box.
In fact it's probably better to just turn off the main.  And it is still wise to proceed with caution, checking and double checking to verify no presence of voltage as you go.