The word that first occurs to me is "courier." Nowadays, a "courier service" is a special, fast service for delivering letters and physical message. It implies that someone makes a special trip to carry a single letter directly from the sender to the receiver--rather than delivering many letters on an organized route. In large cities, we have "bicycle couriers," who carry messages via bicycle, because a bicycle--when ridden with skill--can move faster in traffic than a car can.

I'm no native speaker but from other vocabulary referring to people on horses (like e. g. police) I would assume that he would be called a mounted messenger or a mounted courier.