Make sure that Configure IPv6 is set to Link-Local Only on both WiFi and Ethernet in Network / Advanced on your Mac.
Temporarily, try connecting an Ethernet cable from your Mac to one of the LAN <--> ports on the AirPort router. Restart the Mac and the AirPort and try again. When you open AirPort Utility, click on the Other WiFi Devices tab and make sure that Ethernet is selected if you try the Ethernet connection method.
If no luck, which model of AirPort router do you have?
Is this a new AirPort that you are trying to configure for the first time, or an AirPort that has been working OK on your network in the past?