Hi Christopher,
When you press the upgrade button it initiates a command which takes care of everything including the update script. So really there's nothing to do or that can be done unfortunately.
You could install your old MediaWiki version (http://download.wikimedia.org/mediawiki/) locally, for example using XAMPP (http://www.apachefriends.org/en/xampp.html ).
Then use your saved database, upgrade locally to the current version of MediaWiki, do a database backup of that version (for example with MySQLDumper, http://www.mysqldumper.net) and migrate it to your production site.
hth Frank