Attracive Deck Lighting/Outlets How?
If your inspector is very lenient, and you want "quick" but not necessarily "best" or "safest", you might be able to staple UF to the bottom of the railing overhang. This would be especially true if the railing was high enough off the ground on the outside that it would be unlikely for someone to hit it with something. As was mentioned in another recent thread, protection from physical damage has a wide range of interpretations.
Another option would be to make large notches in the post, put small conduit in the notches, and then cover it with some kind of decorative wood slats or something. How you cover it would be up to you, and if the wire is in conduit the code doesn't really care as long as the conduit is secure and doesn't rattle around in there.
Similarily you could put UF in the notches and then cover it, but if you are going to notch you might as well continue the existing conduit system you have going on the bottom of the deck.