domain name system - Suitable Record Type for a web server using a non-standard port?

I need to set up some servers that must be ran on ports above 5000. (This is a request from the client for this job, due to internal policies, and I can't do anything about it). What is concerning me is that, among other things, I have to set up a web server to run on a port different than 80, but the site must still be able to receive requests like example.com/contact and example.com/register . Since it will not operate on port 80, I won't be able to use A Record. So... is it even possible to do this job? I could configure Apache to use 5080, or whatever they want, but then what? How could people outside the LAN access it by the address example.com/whatever ?