Outdated answer at this point, but just in case others stumble upon this thread. It's not all that uncommon, you'll be fine, even though your neck might feel locked up for a day or two. Just stay active, even if your first instinct is to lie down and wait for it to pass over.
Fitness
Sounds like you pulled a muscle. You can't hurry recovery for that. Pulled muscles hurt when they contract and you might think stretching will help, but it won't.
In the future you need to make sure you're warmed up properly before lifting.
For soreness and recovery I would suggest a massage gun especially for your neck area just use slow setting.
Edit: and making sure you warm up and maybe stretch (yes your neck/back) before or after since it has happened before. And OHP can be finicky with form because there's alot of stabilizing that goes into it.
I'd also recommend a pair of lacrosse balls to use against the floor or a solid wall (careful with plasterboard and such).
Yes those are also great. I'm not sure I would use them on my neck area though.
I've used them on posterior delts and traps to decent effect.
For the neck, a friend who is into rock climbing gave me a little cork 'peanut' looking thing which actually works pretty nicely:
Gotta keep your head from turning even the slightest amount under load.
People aren't going to be able to diagnose you over the internet. Take some time away from lifts that use muscle in the area if you feel pain. After a week or two if things don't start feeling better see a doctor or a DPT.
It could also be a nerve thing. I’ve had that happen with Upright Rows. For me the immediate prevention was loosening my neck (avoid clenching my neck) during neck-adjacent lifts like that. Long term fix was PT to resolve some posture issues I have with my neck that can cause nerve impingement.
I'm no expert but I think the OHP will rely a lot on your own makeup in terms of alignment. If your shoulders are out of alignment at all, I think it makes a difference in terms of injury potential more than other lifts.