In my garage I have two separate overhead light bulb sockets, each is connected to a separate wall switch. One works, the other does not. Using a tester, there is no power at the socket that is not working. I have replaced the wall switch and the fixture, it still does not work. The last thing I can replace is the circuit breaker in the electrical panel.

My home was built in the 1960s and the original electrical panel is still in use. As you can see from the photo, there is not main breaker to cut power to the panel. How do I replace the circuit breaker without cutting power to the panel?

Thanks
CH

How do you know the breaker needs replacing? A bad breaker is a rare thing.
I bet the problem is elsewhere.

I would use the tester at wall switches, It is hot there, be sure hot goes to other switch,
Then check other side of switch goes live when switched on.
Breakers can trip but not throw, lite pressure or resetting breakers can solve, Then check voltage out of all breakers, If you find one in the ON condition and no power then replace.
An electrician will turn off braker, remove wire and cap or don't make contact. (Some panels are wired wrong/fed from 2 breakers by accident, even though breaker is off wire is still live) carefully pivot breaker from the center out, best to try on dead panel or at supply house, don't touch the sides of the Panel while doing this, Stay Isolated, no Bare Feet, Rubber shoes work well.

Agree with Stanfortyman that it is highly unlikely to be the breaker. To be the breaker there would have to be other things that do not work or this one light would have to be the only thing on the breaker. Did you check to see if you had power at the switch when you changed it. Turn off the breaker, then check to see what else does not work in order to trace wiring, logically if not physically. Then begin looking for faults at these points.

Where is the pic?