I've installed the MS Outlook app on my Xperia X Compact and all works will with Outlook.com apart from the Contacts app sync.
I can see the contacts stored in my Android Contacts app in Outlook but when I add a new contact using the Android Contacts app the new contact does not sync to Outlook.com (as it's configured to do). There is no option in the MS Outlook app to add contacts so I have to use the Android Contacts app.
When the next contacts sync from Outlook.com to my phone runs any new contacts added using the Android Contacts app are removed. The only way I can add new contacts (that remain in place) is to add the contacts into Outlook.com itself and let the new contacts sync to my Android contacts.
Why doesn't the Android Contacts app sync new contacts into Outlook.com?
You can try using a workaround with switching to Google contacts suggested by Tahakaro. That's probably the fastest way to check where's the problem. Another workaround is Akruto. It can sync all Outlook data with Android over Wi-Fi. Here's a link: