You're getting the 415 error of Unsupported Media Type when Apigility cannot deserialize the data coming from the client. This recently was called out in the documentation.
I suspect your problem is due to the content-type being sent from postman. Pay special attention to the Content-Type Whitelist listed for the service and make sure it contains the content-type you are sending.
For example, if your service has only has application/json in the Content-Type Whitelist and you send the PUT/PATCH with postman as x-www-form-urlencoded, you will get a 415 error of Unsupported Media Type. If you change postman to send the PUT/PATCH with a content-type of application/json and the request body contains valid JSON, Apigility should accept the request.
You can check the content-type postman is sending by clicking on the "Preview" button just to the right of the "Send" button.