The problem is that Apple refuse to build upnp into their routers.. they use their own system called NAT-PMP which is not compatible..
End result you either do all manual port forwarding or you buy a standard upnp router and use the airport for wireless access point.
There are heaps and heaps of threads around here..
eg PlayStation 4 and Port Forwarding
Just do a search for whichever PS you have and port forwarding or mapping (apple use the later term).
The method of mapping is in airports is here.
AirPort - Port Mapping Basics using AirPort Utility v6.x
Now to be sure I understand you .. are you using two playstations in the same house and are trying to play against each other directly rather than using Internet live type play?? Local lan issue can be different as ports are all open and it can just be the airport is failing to register the protocol being used.
It is a lot easier to buy a standard router with upnp as soon as you leave the apple walled garden and want to use anything but apple products you are not catered for.