Its natural and correct, although the meaning is just a little bit different. It doesn't imply the person you're talking to is going to do as much work as if he were actually to introduce you and the man. it's more like he's just going to arrange for it to happen.

I think "Are you going to introduce him to me today?" is the most natural. You can also ask, "Are you going to have me meet him today?" I think "me and him meet" does not sound natural.