I've built a basic facebook messenger bot with PHP. How do I enable timed push notifications?
For example, here's an appointment booking scenario:
This is a broad question and has little to do with bots themselves. Think of bot as only a communication protocol between your server and user. As you've mentioned, store user ids/meeting times in a database.
Then have a cron running, calling a link on yuour server which check the appropriate whether should a user be sent a notification now. Then send him a message and mark this action in a database, so it is not handled twice.