I think you would be better off either using native web push notifications OR using something like Pusher or the alternatives Laravel Websockets for this and then you can push real-time to specific channels.
Again if your just doing all this for one popup and only ever one, the top two may be overkill.
If you want to do it using your example, the easiest method is setting a a notification table, where you set the message to the user, and on page load, you have a partial that checks for any new notifications (provided in a global view composer) and if there are show them the user and mark them as read, so the next page load they don't appear again.
In the backend you'd trigger an event (i.e. PostApproved) and a listener would be like .. NotifySubmitterOfApproval and in that listener you would create a new notifications table entry for that user.
Personally if I'm going to be using more than just one notification, I'd make use of webpush/sockets ;-)