Light fixture with Pull chain
Reconnecting the light is just white-to-white and black-to-black.  That's pretty simple, but it never hurts to double check.

As for the second question, it does look like the pull chain switch is riveted to the lamp socket.  Those types of fixtures are cheap these days ($15 for fixture, $5 for just the switch), so you should just replace the whole fixture.  Don't worry about finding a pull chain mechanism and trying to splice the wires, etc.

If you really want to keep that fixture because it has sentimental value, you would want to cut the insulated wire going to the pull chain switch as close to the switch as possible to give you plenty of room for a wire splice.  You won't be able to attach directly to the light socket since its riveted, so a wire splice is the only option.