Is it true that I can use the octopus card for free??

No, it is not for free. As you buy the Octopus Card, a certain amount of deposit is in it. Thus when you use it on MTR or bus, you just need to scan your card and the corresponding fare is automatically paid. Similarly, you can also use it to pay for the express fare.

For sake of money saving, the best way for you is to take the hotel free shuttle bus to Causeway Bay Station, from where you can walk to Hysan Place, Hennessy Road within 200m. There you can catch bus no. E11 to the airport directly within around 75min. The bus fare is 21HKD per person.