I can’t seem to get Outlook email categories to sync with my Nine emails.

Categories are not able to be synced via ActiveSync protocol.  It is an ActiveSync protocol limitation. If the emails including categories are synced from server, Nine extracts the categories values separately and then store them in Nine DB.
If you set the categories on emails and the emails are synced to Nine, you can see the categories on Nine.
The category color is unable to be synced as well, so you should change the color on Nine manually.

You can change the category color with the following steps.

• Settings > Click on your account > Categories > Click the color icon and then change the color