There are two ways to handle this, the most common approach is to have your router or firewall run a DDNS client and update your DDNS service, sadly Apple AirPort Extreme is not really suited to this. You would have to check your router/firewall to see what options it has. The other approach is to run a software DDNS client application on your Mac to do the same thing.
DynDNS is one of the oldest providers of such a service and does provide a Mac client, no-ip.com also do but I don't find it as good, there is also dtdns.com which is a bit more interesting as their Mac client is in the form of a command-line script which would make it easier to automate and run on a Mac server even with no users logged in on the server itself.
The Mac server can run one of these clients even if it itself only has an internal private static IP address. I do this myself to access my home Mac. You would have your home router then do port forwarding to the internal private NATed IP address of your Mac server. As a result I can if I wish use my home Mac as a web server via a standard DNS name even though my home Internet IP address is dynamic.
For my place of work I have proper official static public IP addresses as I run multiple servers.