Managing personal Outlook.com account settings, security, and privacy
I have Android 4.03 and this wouldn't work. It isn't just a firmware issue, it is a problem with the app which only seems to work when you're on Wifi (not when on 3G/4G/LTE).
There are good instructions how to get rid of the app and go back to the old Hotmail app in the 'most helpful reply' here: http://answers.microsoft.com/en-us/windowslive/forum/mail-sync/why-does-my-phone-app-for-outlookcom-keeps-giving/c52c6772-c73e-4014-adbe-b9a7cff3c041?page=1