Is there a free airport shuttle bus from hong kong airport to the nathan hotel?

No free bus can directly take you from HKIA to your hotel. I'd suggest you take Airport Express to Kowloon Station with the fare of 90HKD. Then change to shuttle bus line K1 and get off at the 6th stop - Eaton Hotel. Then walk to your hotel within 3-5min. Or take bus no. A21 to Nathan Hotel, Nathan Road. The drop-off site is very close to your hotel. In this way, you just need to spend 33HKD, but the duration is almost an hour.