Welcome to the world of running, Jessica!
Arm pain when you first start running is not uncommon. Lifting helps, but no, you don't need to lift more if you already are doing some lifting. What does your form look like? What are your arms doing when you run? What are your hands doing?
When you run you want everything moving forward. Your arms should swing easily from the shoulder in natural arcs, not over-exaggerated or too stiff. For your hands pretend like you are holding a butterfly in your hand. If you clamp your fist shut, you're going to crush the little buddy and we don't want butterfly carnage! That will help you relax your hands. If you feel like you need something to "hang on to" tuck your thumb into your hands and gently hold it.
Your arm swing should be forward, not across your body. If you're crossing your midline, you're using too much energy on arm swing, energy that should be reserved for running.
How about your shoulders? Before you start your run shake your arms out, then lift your shoulders up as HIGH as you can right up to your ears and let them drop. Shake it like a Polaroid picture baby!
Finally, are you running too hard? I have had the crampy feeling when I was trying to go out much faster than I should have. I one time lined up at the start of a 5k with the teenage track stars. I am NOT a teenage track star. I am an adult non-track star who runs an 8 minute mile on a good day. I was running so fast (for me) that my arms cramped up and I felt like I was having a freaking heart attack! I slowed my pace to MY pace and let the track stars leave me in their dust and immediately felt better. A lot of new runners feel like they have to go fast, but that's not so. Running, unless you truly ARE a track star, should be enjoyable. Speed can come AFTER you've built a good physical and mental base. (Let's face it, if it hurts every time you do it, after awhile you won't want to do it!)
Incidentally, I took first in my age group on that race, and ran my fastest 5k ever, even with a well-meaning volunteer who made us all stop at the crosswalk because it SAID "don't walk"! (No, I'm not kidding.) Maybe I should line up with the teenage track stars more often!
