IIRC, the Apple Airport products only operate as wifi-extenders in WDS mode (Wireless Distribution System). There is some requisite setup and I'm not sure if Apple's WDS implementation is compatible with other systems (i.e. OpenWrt or other hardware/firmware combos). WDS also has the disadvantage of severely impacting the available bandwidth (by a factor of 2).
If you have the option to connect the Airport via ethernet to your main OpenWrt router, that is the best option (the wifi setup on the Airport would then be just as a bridge). Otherwise, you may be better off picking up a dedicated wifi extender device or another OpenWrt compatible router or AP.
So it sounds like you're looking for a wireless > wired bridge instead of a wifi extender (which is wifi > wifi).
In that case, if I recall correctly, this can be done by putting the Airport into wireless client mode and connecting to your wifi network. There shouldn't, in theory, be any reason that it won't connect as long as the basics are set correctly (2.4GHz or 5GHz, SSID, correct standard (g, n, ac), security type, and password all correct). Once it connects as a client device, the ethernet ports should become active as bridged with wifi.
Test this within close physical proximity of your OpenWrt router so that you can rule out signal strength problems.