How can I ping a given host behind an Airport Extreme Base Station
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.