I would stay connected to this person if I truly enjoyed interacting with him. But I would not hesitate to inform him that my what my ex is doing doesn't interest me and to please stop the updates. Be honest - tell him what you told us - that you want to put him behind you and move on.
If you don't enjoy interacting with this person, you might want to let him know that you'd prefer to cut off all contact with him, since you no longer have your ex in common. If that's too awkward for you, just ignore him and he'll go away. Or block him from contacting you, if you can.