I missed first step of the stairs coming down backwards landed on my shoulder. It didn't hurt until two days passed. Now severe pain in upper arm. Broke?

Maybe (not) . . . It's impossible for me to tell whether you/that person broke his/her arm or not. It's less likely if pain didn't start until 2d later but perhaps it took that long for adrenaline from fall to wear off, so to speak. Regardless, given onset of severe pain now, it's best for you/him/her to see GP ASAP. Go to nearest A&E if GP not available. I'd also worry about bleeding in brain (head injury) etc.