I think the most important aspect to storing it is keeping it in a cool place and making sure it doesn't get jostled too much. We keep our pens in a cup in the fridge.
I'm not sure why you would have trouble keeping it upright. ProZinc is in a vial. I used to put mine back in the box after each use and put it in the refrigerator. I put it in a spot where it would not get knocked over/out.