Always warm up. Trainers for shot putters suggest light exercises focused on the wrist and hand, also arm exercises and even jogging. Working up to the shot put is a great way to avoid the sudden pressure on muscles that is part of the shot put.
Stretch. The shot put activity acts on a variety of muscles; not just the hands, but the arms, chest and back. Back injuries are common in shot put; other common problems involve tendons and ligaments. For all of these, it is important to stretch, to keep the muscles strong and supple, able to deal with weight and stress.
Try yoga or pilates. Some shot put trainers suggest that these body-therapy exercises are a good way to get stretched and work muscles up to task. Pilates, as an ergonomic, high-tension set of activities, really works your muscles and gets your body ready for a wide range of intensive activity, including the shot put.
Try calisthenics and activities like jumping rope. You may think of the jump rope as a tool for more running-based activities, but trained shot putters suggest the jump rope is practical for them, too. Activities like the jump rope make your body more agile and increase all kinds of stamina and dexterity. How does this help in shot putting? Agility is useful in the "lunge" that occurs on a throw, where much of the body is in play.
Keep the weight in the right place. Trainers suggest the weight of the shot should be mostly on the fingers, with the arm angled to absorb the rest of the weight. For sitting shot putters, the back should be straight, so tension from the throw does not get absorbed by the back. Getting the right balance will help you avoid the problems of tension on muscles that are not commonly used for the throw.