What you’re wanting sounds more like a DNS/Apache problem than a WordPress problem.
You would need to set the DNS alias and vhosts on up on apache in your intranet. Doing it via WordPress is not the way to do it, and would involve jumping through hoops and putting a massive amount of effort in for something that runs slow and has problems. The port 8080 is also going to be an issue here.
The only other method that’s straightforward is to cloak the page and put a full page iframe leading to your local machine, but it is a hack/kludge, and all the URLs will still show your local machines URL not the pretty URL. It will also be decloaked if the user opens a new tab or window, and you won’t be able to copy paste addresses or remember pages between browser sessions
If it worked with WordPress as you desire it to, then I could set up an install, specify apple.com as the site url, and steal all of Apples traffic.