Make dynamic IP address fixed?
Dynamic DNS (DDNS) is what you need if you don't want to pay for a static IP, but as Arjan pointed out DDNS only works for inbound traffic. As I see it you have three options:
Pay Time Warner for a static IP address.
Set up a proxy server that will send its IP address (which will be static) instead of your IP address.
Use third party firmware on a suitable modem and force a static IP address. Not sure what your terms and conditions are with Time Warner, but in all likelihood they will not like this and might cancel your service if they find out you are doing it.
Only your ISP can give you a fixed IP address. You could look to using a VPN to a device on a fixed IP address (for instance a virtual server you managed yourself), but that's about your only other option.
Alternatively you could talk to the admin of the server and see if they can set it up so that you can use a VPN to access the server.
Security by blocking IP-addresses is primitive.
But like Cry Havok wrote, talk to your ISP. Keep in mind that a fixed IP usually costs more than a dynamic one.
If your firewall allows you to set up rules by URL you might want to try out a service like http://www.dyndns.com
Most routers have options for dyndns and such or you can install a client on your machine to update your IP with the dyndns service