If your exchange online is part of Office 365, you should definitively take a look here : http://dev.office.com
You will find many tutorials to get access to your mails.
Take a look here : http://dev.office.com/getting-started/office365apis
It uses the Microsoft Graph to query your mailbox and retrives messages :
https://graph.microsoft.com/v1.0/me/mailFolders/Inbox/messages
The authentication process is provided byt Azure Active Directory, through ADAL SDK.
In Visual Studio, you can create easily an application, and then call the submenu 'Add connected services' and you're done !