Can I use my octopus card from Mongkok MRT Station to Airport?

You can take metros by your Octupos Card. You can take Tsuen Wan Line at Mong Kok Station to Lai King Station and take Tung Chung Line to Tsing Yi Station. Then you may transfer Airport Metro line to the Airport directly. The whole trip will take about 55 minutes at the cost of HK$60/person.
Besides, you can also walk about 200 meters westwards along Argyle Street and turn right to Relamation Street for 50 meters to Fife Street, Reclamation Street Bus Stop. Then you can take bus no.E21 to Cheong Tat Road Bus Stop. Upon arrival, you can see the airport directly. This trip will take about 75 minutes at the cost of HK$14/person.