Off the top of my head:

1. Talk to the DNS manager to create an A DNS record for example.mysite.com pointing to the IP address of your Apache server
2. Enable the NameVirtualHost directive in httpd.conf (it could be in an include file, so dig deeper!) - Just ensure you use
    name-based virtualhosts.
3. You will need to ensure that your Apache instance listens on port 8081 as well. Make an additional Listen rule .
    When you say localhost, then obviously this example.mysite.com is running on your Desktop, right? Because if it's not, then you will NOT be able to access it.
3. Create a virtual host with the name, i.e.
    <Virtualhost 127.0.0.1:8081>
         ServerName example.mysite.com
         DocumentRoot /some/path/to/your/files
         ErrorLog /some/path/to/example-error.log
         <Directory /some/path/to/your/files/>
            Order allow,deny
            Allow from all
         </Directory>
    </VirtualHost>

You may need to edit /etc/hosts to add an entry for example.mysite.com mapped to 127.0.0.1

Restart Apache instace (apachectl graceful) and run tests. Look for clues on example-error.log

PS: This assumes a monolithic configuration of Apache, but with include(d) config files..

E&OE :-)