It depends on what this system is you want to shut down.
You will need a router with dynamic DNS(most do) and a DNS account(you should be able to find something free if you look hard enough, dyndns.org doesn't offer free accounts anymore). You will use a static IP for your ethernet shield, forward the necessary port in your router settings, and put your DNS info in the router. At this point you should have access to your arduino from any internet connection. Making a simple page with links that make the arduino do stuff should just be a matter of playing with it until you get something that works for you.
If you use the arduino nano you will need to interface with the ethernet shield with a breadboard.
So obviously there's more detail to it. But those are the things you need.