I am not familiar with the Airport base station, Looking at the device specs it supports 6to4 tunnelling, This would require the Airport to have a public IP address for the tunnel to work (http://en.wikipedia.org/wiki/6to4)
If you are unable to give the Airport a public IP address then you could setup a tunnel with an IPv6 tunnel broker (http://en.wikipedia.org/wiki/Tunnel_broker) and use stateless auto configuration to give out IPv6 addresses on your network.
Most tunnel brokers (if not all) run the service for free.
You may need be able to configure this on your router, if not then you could setup a small server to do this.
I have used Hurricane Electric's tunnel service in the past to do this http://www.tunnelbroker.net
They also have example configuration for various devices and operating systems.
Edit: I just had a quick look at Hurricane Electric's website and they have example settings how to setup a manual tunnel using an Airport (The example configuration are viewable after you have created the tunnel), So it looks like you could set it up directly on the Airport.