I have a paid web host on godaddy, I would like to know if it is possible that I use a custom port to redirect to a specific folder on the server (ex: a web mail server or a private folder)
If it's a dedicated server and you have full root access (or at least access to do as you wish with apache), then sure, this should be quite simple.
Say, for instance, you wanted to associate port 8888 with a certain directory...in your main apache config file (apache.conf or httpd.conf), add:
Listen 8888 NameVirtualHost *:8888
Then add a VirtualHost:
<Virtualhost *:8080> ServerName foo.example.com DocumentRoot /path/to/directory </VirtualHost>
I haven't tested this, so I'm not sure it's 100% correct, but it's very close, and you should be able to adapt it to your environment.