An alternative is to look at Office 365 groups or Microsoft teams and use the native mail alerts and messages there.
If you need further flexibility you could convert the shared mailbox to a user mailbox. Use that login to create a flow or Powerapps with that account and have it trigger alerts and send to your primary mailbox.
Sticking with just the shared mailbox I think you are limited without some custom code solutions, but one thing that may be worth investigating is maybe using transport rules to create notification for deliveries. It probably won't be exactly what you are looking for but may give you more options than what you will have with Outlook rules only