1. Buy some hosting outside the university and migrate your site there.
OR
2. Speak to your IT department and migrate onto a proper server there.
There's lots of security issues etc. when running a server on the public internet that you probably don't want to have to worry about.
Set up an external ip that accesses the server that your site is on. Create local and external DNS records giving the site a name.
In your moodle folder, change config.php file to reflect the dns name of the site.
Go to "yoursite.com"/admin/tool/replace/index.php and run the replace tool to update the database with the new dns name (backup database first!).
Then you should be good to go!