Let me see if I understand your question. You want the requests that are received via site.ourdomain.com to be routed to site.hosteddomain.com without changing the URL on the client browser, is that right?
If so, all you need to do is to configure ARR as a reverse proxy. The UI for this configure differs slightly between ARRv1 and ARRv2. If you can tell me which one you have, I can provide you with the exact steps.