Surface Mount Ceiling Fan with Conduit to a Half Hot plug
This is precisely what surface raceway is for

What you want to do is use surface raceway to bring the switched hot, neutral, and ground from the half-hot outlet to the new fan location.  You'll need to put a compatible extension box (starter box) on the outlet to provide room for the connections, and use a raceway system (Legrand's B-series metal raceways work) that has a fan-rated box (B4F/BW4F in the Legrand B-series) available.  As to wiring inside the raceway, you can run a NM cable or single THHN wires.

(As a sidenote, EMT with standard boxes and fittings also works for running exposed wiring safely, but you may have trouble finding a surface mountable fan-rated box outside of a surface raceway system.)