Fastest way to fetch specific commit from (outdated) full clone?

Question

I have a full clone of a repo, which is slightly outdated.

# A few days ago:
git clone url/foo.git

Now I'd like to fetch and checkout a new commit which is not part of the cloned repository yet.

I can do:

git fetch && git checkout 409b7405

I assume running this will be faster as it downloads less data:

git fetch 409b7405 && git checkout 409b7405

Will it be even faster when fetching with --depth=1?

git fetch --depth=1 409b7405 && git checkout 409b7405

Notes:

The use case is running something on a temporary cloud server instance, so everything will be deleted after running the job, and no further fetches will ever happen on the repository.
I'm really just worried about total time, so I'm wondering if --depth=1 might need more CPU on server or client side to run, negating any network saving (in case of a fast network).