Phlebotomist missed vein is hit a nerve. now have shooting/shock type pain down forearm, back of hand when moving certain ways/lifting objects. does this heal on its own? if so, how long does it take? should i avoid moving that arm for now?

It may be a tendon and not a nerve that was hit. That is most likely. It should gradually go away. Heat, ice and anti-inflammatories my help settle quicker. No need to avoid things. If it is a tendon, it should take a couple of weeks, if it is a nerve, may take a couple of months to completely go away.

Assuming a nerve was contacted or compressed by a local hematoma, maximal symptoms accelerate over 24-36 hours, and thereafter, if myelin covering was affected, the typical recovery is ongoing within 3-4 weeks. Permanent injury to the axon is rather unusual and not expected but if this did occur, several months of symptoms are to be expected.