by default you shouldn't need to manually open/map any INCOMMING ports for full SL functionality (although you can specify which port SL should use for more complicated LAN setups).
but SL does assume that is has full tcp & udp OUTGOING access.
virtually all domestic/light-commercial dsl/cable modem/routers allow all outgoing connections by default, including the Airport Exp/Ext.
however you might have a software firewall installed that is blocking SL - there might be an application white-list that SL needs to be added to. this would certainly explain your symptoms. although you'd normally get a pop-up from the firewall to ask if it's OK to allow SL, unless you already accidentally said No / Always No.
if you're running SL on a Mac (do you get the frequent freezes/spinning-beach-ball too?), the OS X firewall doesn't block outgoing traffic at all. all you can do is white/black-list incoming connections (all/essential/selected-apps).
obviously if you're running both a hardware & software firewall, for any app that does need incoming ports mapped, you have to configure that in both - unless you've got UPnP or NAT-PMP enabled so that apps, OS & router can work it all out for themselves.
as for port-mapping on an Airport, in case you ever need to do it:
remember that the Airport Express/Extreme can be configured to be merely a wifi access point (where your dsl/cable-modem/router does the routing/firewall functionality, so port-mapping should be done in there, not in the AirportE), just as easily as where the Airport does all routing/firewall/wifi functionality, and the dsl/cable modem/router is used in bridged mode. same thing for one of the other replies, his Airport is only config'd to extend the wifi network of another base-station, not do routing/firewalling.
so first up you need to know which way your Airport is configd. launch the Airport Utility and click on 'Manual config...'. click on the 'Internet' icon, & note the 'Connect using' & 'Connection sharing' pull-down lists. the various combinations of those are what set the Airport's fundamental config. check Airport doco for details...
maybe one day Apple will turn those screens into something with pretty diagrams so that "normal people" can actually understand what's going on and stop wasting time trying to configure their Airport when it should be their dsl/cable-modem/router they should be changing

(or vice-versa).
cheers, Anthony.