Maybe try altering your running style a bit to reduce excessive lateral movements. As you run, periodically shake your arms and shrug the shoulders to loosen any tension. Moving the head from side to side (taking ear towards shoulder) helps too.
Weights will help to strengthen the upper body. Do military presses, lateral raises and upright rows using dumbbells for the shoulder area. For back strength, do rows and the lat pull down machine.
Agree about core strength - also never underestimate the value of full press-ups for strength. Learn how to do them and you'll never have a weak upper bod again!