How can I temporarily fix a leaky faucet?
Other than turning the water off at a location before the faucet, there is no real temporary fix - you either fix it, or you don't. 

The problem with really old faucets is finding replacement parts.  I had one from the '70s with a similar issue and I couldn't even get replacement parts. I opted to just replace it.

That being said, replacing it should not be a huge job.  I was able to replace an old 3 handle faucet with a single handle faucet in a couple hours (my first real plmbing project). They make special remodel wall plates specifically for this that allow you to cut a large enough hole in the wall to replace the faucet, and then cover it up with the plate:



If you are able to access the wall behind the shower then it is even easier since you can easily cut and repair a large drywall hole.

If you are handy you can definitely do it yourself, or call in a plumber to do it for you.  I imagine a plumber would make quick work of this.