The recommendation is to use the “Page Viewer” web part to help display your Outlook Web App information within SharePoint Online.

To add the “Page Viewer” web part on a particular page:

• Go to the page you want to add the web part to
• In the ribbon click on the following Edit <tab> | Insert <tab> | Web Part <button>
• This will bring up the web part adder – select the “Media and Content” <category>
  • This will bring up a list of web parts – select “Page Viewer”  <web part>
  • Click add <button>

To then view Outlook Web App segments in the web part:

• Open the new Page Viewer web part’s tool pane or click down arrow | Edit Web Part <menu item>
• By default this is set to view a Web Page
• In the “Link” <edit box> enter in your Outlook Web App URL:
  • HTTP://technet.microsoft.com/en-us/library/bb232199.aspx - use this article to see how to construct the URL for simple inbox viewing, to even more targeted viewing of specific folders. The options are in the “Syntax” section (which needs to be expanded to see it); here you’ll learn how to configure the URL to pull in inbox, contacts, calendar, and tasks.
• The URL of Outlook Web App web part in Exchange Online must have an exsvurl=1 in the webpart URL in order to make web part work:
  • The basic URL construct: https://<server name>.outlook.com/owa/?exsvurl=1&cmd=contents
    • You can go to your Office 365 portal home page here: https://portal.microsoftonline.com/IWDefault.aspx
    • And then click on “Outlook” at the top middle of the page. This brings you to your personal Outlook Web App where you can determine your server name to help build out your preferred URL to plug into the “Page Viewer” web part. 
  • For example in my beta account, my URL would be: https://pod51008.outlook.com/owa/?exsvurl=1&cmd=contents
  • Another examples if I want to view a weekly view of my calendar: https://pod51008.outlook.com/owa/?exsvurl=1&cmd=contents&f=Calendar&view=Weekly
  • Or to show my tasks: https://pod51008.outlook.com/owa/?exsvurl=1&cmd=contents&f=tasks