If you see that the Inbox is not updating automatically, please submit an In App ticket (Help, Support). The In App Support Team raised this as a top priority with the Product Team and a lot of work has been done to address the issue.
Some of the latest changes include:
Fixed issues that were delaying reconnect, for example after coming out of Sleep Mode on the computer
Updated sync prioritization to sync inbox first
Reload to get fresh data from the service more quickly
If there are still issues, then submitting a ticket helps get it investigated so any new fixes can be implemented. For example, if you are seeing dramatic difference receiving on your mobile device compared to New Outlook.