MacBook Air audio input/output, mic/headset
The port does support headphone+mic combos, which you can verify for yourself by plugging in any headset that's compatible with an iPhone. Apple notes this on the Audio section of the Tech Specs.

The reviews on the product page suggest that it works with other Macs, so it seems likely you've got defective hardware somewhere in the chain. Try testing the components individually with other known-good devices. Test the adapter + headset with an iPhone or iPad, the headset itself with a PC, and the MacBook Air port with an iPhone headset.