@David_B I've done similar to this to my garden office and it works great. Check out the products starting with names tl-wpa and ending with kit.

Your mileage may vary depending on the quality and exact set up of your wiring.

The main challenge I think would be finding a weatherproof socket that could accommodate the wireless unit, and you don't want to use an extension socket. Better to locate a socket inside or at least sheltered.