Your External IP Address (or, as you call it, the IP address reported by whatismyip.com) is not under your control and you can't simply set it to static. Your ISP gives you a new one when you reconnect, although there are chances that you might get the same, you should never count on it.
In order to solve this problem you have two choices:
Some Internet Service Providers (ISPs) offer you a "Static" IP service for a monthly fee. With the lack of IPv4 in the world, this service is unlikely to exist forever to individuals and it's usually reserved for companies and services that really need a static IP. Not all the ISPs offer this, you can always ask.
Use a Dynamic DNS provider. This will mean that although your IP address will eventually change when you disconnect (or your provider wants), you will always be able to get "back" to your Airport's IP by using a "name" rather than an IP. Any computer in your network will have a small piece of soft (some routers can do this automatically, I'm not sure about Airport Extreme, don't think so), that will periodically check your IP and if it has changed, it will inform the Dynamic DNS service that you use so it can update the value.
Some Dynamic DNS Providers (Mostly a free service) are (in no particular order):
Think of this as if instead of having a fixed phone number (static ip), a person could call you by "name" and the phone will check what is your current number (wouldn't that be great? ;) Nobody knows your number, only your name, because your number changes often (your ip!).
You haven't mentioned why you need a Static IP in the first place, but if you can't get it via your ISP, your only hope is to use Dynamic DNS. For what is worth, DDNS works really fine and it's very "fire and forget". Once it works, you no longer have to worry about IPs or anything, you just connect to: XXXXX.no-ip.org (for example)
You can make this host the default host in your AEBS settings, also known as the DMZ host.
Note however that everything that isn't mapped otherwise will be routed to that host, effectively making that host public from the internet. This might make this host more vulnerable for break-in attempts, be sure to lock all services down properly, and consider putting on a firewall.
Depending on your version of the Airport Utility client, you'll find the setting here:
AirPort Utility 6:
AirPort Utility 5.6:
Overall, I wouldn't recommend this approach because of the security risks on the one hand, on the other hand it doesn't look like a very reliable approach. If the host doesn't respond to pings there's no reason the host can't still be up, and if it does respond, are you sure it is the host responding, and not for example the router after a config reset?
I'd consider polling a dedicated service like httpd which you run on the host and you forward properly from the AEBS. You could run it on a non-standard port as well. With some scripting you could also return a whole lot more detailed information about the availability of the host and it's services.
Edit:
Since you mentioned the host in question is an ATM, I'd be even less inclined to make it the DMZ host. Consider polling it from another host on the local network and notifying you from there.
Best Answer
What you could try is set the DHCP range to start at +1 and manually set the IPs on your server. For example, set the DHCP range to be 10.0.1.3 to 10.0.1.200. The 10.0.1.2 IP will be free to be used by your server. In AirPort Utility for me this is at Network > Network Options.