There are some good tools available from Microsoft for troubleshooting Exchange Online issues. Below are a summary of the tools that our customers find to be most useful:
To troubleshoot common mail flow issues in Office 365, use the Mail Flow Guided Walkthrough. The walkthrough helps Office 365 administrators to diagnose and resolve issues that are related to mail flow in various scenarios.
The Outlook Connectivity Guided Walkthrough helps you to isolate and resolve any connectivity or performance issues that you are experiencing when you connect your Microsoft Outlook client to an Office 365 mailbox.
The Hybrid Environment Free/Busy Troubleshooter assists in troubleshooting free/busy issues that may occur in a hybrid deployment of an on-premises Microsoft Exchange Server and Microsoft Exchange Online in Office 365.
The Hybrid Migration Troubleshooter is designed to assist when you are having issues moving a mailbox from an on-premises Exchange environment to Exchange Online (On-Boarding). This will also assist you if you are moving a mailbox back on-premises from Exchange Online (Off-boarding).
The Remote Connectivity Analyzer with Message Header Analyzer – This tool is great for troubleshooting a number of Exchange issues. Now, you can analyze message headers to determine what causes mail delays. Learn more here.
Mail protection reports for Office 365 provides you with a detailed view into the email protection data that is available in the Office 365 Reporting dashboard. This workbook is available to all Exchange Online and Exchange Online Protection customers.
The Exchange Online Message Trace Tool allows you to trace messages sent within the past 90 days.
There are some great video overviews of all of the available tools on Channel 9. You can view them here.
A complete list of all of the diagnostic tools available for Office 365 can be accessed here.