As far as I am aware, you cannot.
You may be able to configure DNS records, but DNS only deals with domain<>address translations and not ports.
The port is part of the URL used to access a site or system, eg.www.website.com:8080
People either have to use that or you need a redirect...
There is no reason you cannot do it yourself with a minimal PHP script on the main (port 80) website; just do a 302 (or 307 or 301) redirect to the domain:8080
That should transparently route anyone visiting the site to the alternate version on 8080.
Note that port 8080 is commonly used for proxies and may not be remotely accessible on any particular shared hosting system.
Example redirect script - just change the 301 & URL to whatever you want:
http://www.phpjunkyard.com/tutorials/php-redirect.php
Redirect codes:
https://en.wikipedia.org/wiki/URL_redirection#HTTP_status_codes_3xx