Apache redirect to another port

You should leave out the domain http://example.com in ProxyPass and ProxyPassReverse and leave it as /. Additionally, you need to leave the / at the end of example/ to where it is redirecting. Also, I had some trouble with http://example.com vs. http://www.example.com – only the www worked until I made the ServerName www.example.com, and the ServerAlias example.com. Give the following a go.

<VirtualHost *:80> 
  ProxyPreserveHost On
  ProxyRequests Off
  ServerName www.example.com
  ServerAlias example.com
  ProxyPass / http://localhost:8080/example/
  ProxyPassReverse / http://localhost:8080/example/
</VirtualHost> 

After you make these changes, add the needed modules and restart apache

sudo a2enmod proxy && sudo a2enmod proxy_http && sudo service apache2 restart