I have two local git repositories, both pointing to the same remote repository.
In one git repository, if I do git format-patch 1, how can I apply that patch to the other repository?
Note: You can first preview what your patch will do:
First the stats:
git apply --stat a_file.patch
Then a dry run to detect errors:
git apply --check a_file.patch
Finally, you can use git am to apply your patch as a commit. This also allows you to sign off an applied patch.
This can be useful for later reference.
git am --signoff < a_file.patch 
See an example in this article:
In your git log, you’ll find that the commit messages contain a “Signed-off-by” tag. This tag will be read by Github and others to provide useful info about how the commit ended up in the code.

First you should take a note about difference between git am and git apply
When you are using git am you usually wanna to apply many patches. Thus should use:
git am *.patch
or just:
git am
Git will find patches automatically and apply them in order ;-)
UPD
Here you can find how to generate such patches