That can be done through a premium addon, Patron Plugin Pro. You can lock posts or entire post types with ‘Show after X days’ method, or, ‘Lock last X posts’ method. Or ‘show after a certain date’.
The first 2 would apply to your case of course. Locking posts to show after 30 days would make every post public when it is older than 30 days. Lock last X posts would only lock your last, say, 10 posts. When you post one more, the oldest locked post goes public etc.