What is the cost from Hong Kong Airport to Mongkok? Do I need to buy mtr ticket or Octopus card?

So you prefer the mtr? If you will not travel a lot by mtr, the MTR ticket will be ok. If not, you can buy the Airport Express Travel Pass, which saves more than the standard Octopus. All these ticket can be bought at the airprot upon arrival.
From airport, you can take airport express to Tsing Yi, change to Tung Chung Line to Lai King, then take Tsuen Wan Line to Mong Kok. It altogether takes about HKD70 and duration is about 40 minutes.