If you've been inactive for more than 10 days, Outlook stops syncing the mailbox until your next session for performance reasons. It may take a moment to sync, so you may briefly see your messages from 10 days ago as the app is syncing.
If you are inactive for 30 days, you will see the following text:
Please sign in
Your accounts have been logged out due to inactivity.
Sign back into your account and everything will work as intended. If you have questions, don't hesitate to contact support at Settings > Help & Feedback > Contact Support.