I have the exact same problem. I have cervical spondylosis and stenosis and I can't do any kind of exercise involving my arms, e.g. push-ups, without causing my blood pressure to spike immediately and to generally stay elevated for hours; sometimes for 24 hours.

Doing just a few push-ups can take my BP from 130/80 to 170/100, and this is always accompanied by anxiety, light-headedness, worsening tinnitus and insomnia. Exercises like sit-ups also have the same effect. It really sucks.

As I see it, something about certain movements seems to either be activating the sympathetic nervous system or inhibiting the parasympathetic nervous system.

I've consulted several doctors about this to no avail; they're generally dismissive and/or baffled. Clonidine seems to help and some beta-blockers e.g. Carvedilol might help.