In the past, Outlook or Hotmail app will also sync your contact list. However, it seems that now a different way of sync contacts is necessary.
Here's something I recently found from Microsoft, yet with some modification...to manually setup the synchronization:
On your device, tap Email (or go to Settings), then tap Add account.
Enter your Outlook.com email address and password, then tap Manual setup.
Under What type of account is this? tap Exchange (or Corporate).
If there is a Domain field, leave it blank. If there is a Username field, enter your email address.
Enter your password (or, if 2-step verification has started, you have log in your email account and create an app password from your account management).
For Server name enter s.outlook.com.
Make sure the Use secure connection (SSL) checkbox is checked, then tap Next.
Choose your account options and tap Next.
To finish setup, tap Done.