Is there a PoE (Power over Ethernet) kit available for the 802.11ac Airport Extreme? I.e., a kit that will allow me to power the Airport Extreme from PoE instead of from a wall outlet?
No - the power supply expects 100-240 VAC and lists the current draw as 1.5 A which seems quite high for the heat it puts out, but perhaps with several USB drives connected and max power it gets even half that usage in real life.
You certainly can take DC power from POE using a splitter, but those all seem to provide 1A at 12 VDC which is a far cry short of the wattage listed for the Airport Extreme. I doubt you'd find a DC to AC inverter that would generate enough AC to run the Airport though...
There is a solution, see http://find-a-poe.com/wp-content/uploads/WS-GPOE-12v36w1.pdf - works up to 328 ft from a power source.