I apologize in advance if this question is off-topic/inappropriate, but
the version of org-mode packaged with emacs 22.1 for Windows is 4.67,
while the latest current version is 5.13. I can't seem to figure out how
to update the org-mode package in Windows. Any help?

Michael,

1. Download http://orgmode.org/org-5.13d.zip
2. Extract it to your home directory, in a folder such as .elisp/org
3. Add the following to your .emacs before any other org customizations. Change the directory to match where you put org.
(add-to-list 'load-path "~/.elisp/org")
4. I think that's it.

Good luck,
Scott