I have fibromyalgia in my shoulder. sometimes when i pick up a cold drink with same arm, the pain comes from my shoulder down my arm to my hand, why?

Fibromyalgia: is generalized pain for no reason. It cannot follow a standard pattern for something actually broken/abnormal. Pain in the pathway of a nerve that very often gets pinched such as a nerve in the neck with pain down shoulder to hand, would not fit with fibromyalgia. Xrays of the neck or MRI (better) would show a pinched nerve Without an actual examination, I cannot give specific medical advice.

Are you sure?: Make sure you it's not a problem with your cervical spine. MRI of the spine will help determine that.