How do I do a minor grout repair properly?
When I've done grout repair I've cleaned out the old grout, did some slight prodding to see if anything else was loose, made sure the area was bone dry and then got a small batch of grout from the local hardware store. You can probably buy a small tub of the premixed grout (unless you have colored grout then you're stuck buying a larger tub usually).

I just put it on as normal grout, slap it on, push it in, smooth it out a bit, let it dry and wipe off the excess. It'll take a little while to fade in but now it looks like it was part of the original tile job.