Well, I'm neither a physicist nor a doctor but I'd blame it on inertia (Newton's fault) and I'd probably go see a bone doc if it doesn't get better with a couple weeks rest. Lots of things it could be, sprain, strain, tear, cartilage, nerve, bone etc. In general surfing will put a lot more stress on your muscles and joints than touring or even racing because you tend to go from 0 to 100 instantly every time you try to catch a wave. Huge amounts of stress in the system, weak links break. I find many SUP paddles, blades and shafts, are very stiff and unyielding. Unfortunately your body can take a beating from that. Think about getting a small blade and a shaft with some flexibility. Most if not all makers offer different sized blades but as far as I know most don't offer a variety of shaft flexes. The only one I know of is www.KeNalu.com
Surf technique will be a factor too. If you find you have to really hammer hard and long (>6-8 strokes) to catch a wave you are probably out of position and maybe shouldn't be going for that wave. Watch guys (and gals) who really know what they are doing- they are always in position, they take maybe 2 or 3 easy strokes and the wave picks them up and they are off, easy peezy.
Finally check your paddling technique to make sure you are not bending your lower arm (contracting the muscles you've reported hurting). Muscular force should be coming from your core not your arm muscles. You should be bending at the waist not the elbows. Good luck!