You could say the person jostled you. But "jostle" is a fairly unusual word and not that common in conversation. Another choice is, "He brushed by/past me," which is less strong. If it is very crowded you can say, "He squeezed past/by me." If the action is stronger, close to a push, "He elbowed by me." I think "shove" is as strong as push. "Nudge" often means to touch someone to get their attention.

Depending in the intensity of the action. It would be a brush or a nudge if he moved in front of you without jostling you. But, if the action was rude and you were jarred. I would say he shoved his way through.

You could also use "gently brushed" or "softly nudged". Even "gentle push" would get the right idea across.

He gently touched my arm and moved ahead.