I just tried running curl on it now and it came back fine without redirecting me.

For a private repo you just need to set the correct options to curl.

curl -Lk --cookie "user_session=your_session_cookie_here" https://github.com/your_org/your_project/pull/123.patch

And what is the unix tool I use to automatically retrieve the session cookie and paste it there?

Naturally, you use curl. You just instruct curl to save the cookies that the login process sends back to you. Then you reuse those cookies on the subsequent request.

Found an example here.

https://gist.github.com/d48/3501047

The example link above works for me, interestingly enough.