What would cause sharp pains in my upper arm every time a drink cold liquids?

Referred pain. The cold liquid is irritating the lining to the esophagus/stomach and it is causing a referred pain into the arm. Similar to a heart attack causing pain in the arm. If you have always had this, i wouldn't be concerned. If this is relatively new, you actually might have stomach ulcerations and/or esophageal irritation - again.. If new in onset - get it checked by a GI specialist.