No matter if from 深圳 Shenzhen, 九龍 Kowloon or 香港 Kong, the cheapest option to 赤鱲角 Chek Lap Kok is a combination of 港鐵 MTR and public bus, costing a total of 23.3 HK$ from the Island, and 16.1 HK$ from Kowloon (荔景 Lai King), an unbeatable 2 US$.
From Tung Chung, take bus S1 right from opposite Exit B (3.5 HK$). However, the last S1 bus leaves Tung Chung at midnight everyday, and around 00:20 from the Airport.
If you find yourself at Tung Chung station after the last S1 has left, you will still catch the
Long Win Bus N64, which is a 24h service on a similar route as the S1 (though slightly longer and with 5.2 HK$ more expensive).
However, at night the night busses also provide convenient and affordable options to get to the airport. The N11 for example leaves every 30 minutes from the Macau Ferry Terminal via Causeway bay between 0:20 and 4:50 and costs 31 HK$. The N21 costs only 23 HK$ and leaves every 20 minutes from the 尖沙咀 Sha Tsui Ferry between 1:30 and 5:10.
Other night buses to the airport include the N31 from 荃灣 Wan, the N42 from 馬鞍山Ma On Shan and the N42A from 粉嶺 Fanling.
MTR Journey planner: http://mtr.com.hk/eng/homepage/cust_index.html
Bus route inquiry CityBus: http://www.nwstbus.com.hk/routes/airport-bus/route/index.aspx
Bus route inquiry KMB: http://www.kmb.hk/en/services/search.html
List of all bus routes in Hong Kong: http://en.wikipedia.org/wiki/List_of_bus_routes_in_Hong_Kong