Is there free shuttle bus from HKIA to eaton hotel and how do we go about finding where to take?

There is no free shuttle bus for you take. You can take bus no. A21 at Airport (Ground Transportation Centre), and get off at Nathan Hotel, Nathan Road. After arriving, walk to your hotel directly. It takes around an hour on the way. And the ticket fare is 33 HKD per adult. Or take Airport Express train from HKIA to Kowloon first and then transfer to free bus line K1 to reach there.