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 --keep-cr --signoff < a_file.patch 

As noted by riverofwind in the comments:

Don't forget if you have autocrlf=false for Windows only development you'll need to pass --keep-cr to am to keep those CRLFs

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.