All you have to do is add an account. You do not have to add the android outlook app. If you just add your MS account using defaults you will end up with an imap mail account which is no good.
Add your MS account with a manual setup option to get it setup as an exchange account. You will still not get all three, email, calendar and contacts. Remember, you have to set your contacts to use your MS account.
I tried changing the server type with no success till the end. Can't even remember what it was at the start. I tried s.outlook.com, outlook.office365.com, m.outlook.com. Got email and calendar to work with one or two of these, the exchange account would be there but no sync.
Finally got all three, email, calendar and contacts to work on the built in android apps with m.hotmail.com set as the server. Depending on your MS account you may have to use m.live.com or something similar.
With your MS account already added as an exchange account you can change server types. Go to settings, accounts, tap your exchange account, account settings, tap your MS account again, at the bottom you can change server settings. On this page you will also see boxes to check for email, contacts and calendar sync. You do not have to power cycle the phone each time you make a change.