How to properly introduce a rooster to my 4 month old chickens??

I recommend keeping them separated with some chicken wire or similar so they can easily see and communicate with one another but no touching for the first few days to make sure no one gets injured as they attempt to integrate a new member. Its called PECKING order for a reason. Best Wishes.

 

Can you offer him nighttime and/or weather shelter in the portion of run?
A few days to a week of see don't touch introduction would be good.

Then let him out into the run, and to range, for the first time late in the day, say an hour or so before sunset.
This will test the waters for a short time before they all want to go to roost,
and you can be sure he will not go too far before he's compelled to find shelter for the night.
He may go into the coop with the others, or he may go back to his own shelter.