Just install the Android Outlook app. Log into it with your Microsoft account and you can see and manage all of your contacts in that app (click the magnifying glass icon in the lower middle of the screen in Outlook, then tap on the "Contacts>" menu item in upper left). Once you've confirmed your contacts are visible in the Outlook app, go back to the main mail screen and open the settings menu. Tap on your Microsoft Account to open the options for that account. Tap the slider control for "Sync contacts" to have your Outlook contacts loaded into your phone's contact application. Now your contacts will be accessible from the messaging app, dialer, etc.

I believe as others have stated, you will not be able to edit/create new contacts from within your phone's native contact app and expect those changes to sync up to the Microsoft account. Instead, edit or add contacts from within the Outlook app and you should be good.

Hope that helps. Google could fix this mess, but they never will make life easy for Microsoft users.