Shot put will not mess your arm up. The "throwing" motion used is not dangerous on the arm, as it is "pushed" and does not involve much connective tissue. Rather than segmentation of body parts in the arm, it moves in unison. This is less efficient, but much safer, since you would suffer injury very quickly to connective tissue if you used a traditional baseball throwing motion to throw an 8-12 lb. shot put!
As for weightlifting, I'm a fan of full-body compound lifts that build total body strength and stability. As long as you are not training one side of your body unevenly, it probably can't hurt.