Is there a direct bus from Tong Chung station to Chung King Mansion in TST?

Yes, there is. Walk from Tung Chung Station to Tung Chung Crescent, Shun Tung Road to take bus no. E21X to New World Centre, Salisbury Road, and then walk to Chung King Mansions.

Aside from bus, you can also MTR. Take Tung Chung line at Tung Chung Station to Nam Cheong, where transfer to West Rail line and get off at East TST. Then you can walk to the Chung King Mansions.