I want to go to Hangzhou airport from the train station.

Oh, you can find airport shuttle bus at the railway station. The pick-up station is located inside the long-distance bus passenger station of the railway station (Under Qing-Qin Overpass). You need to walk north along South Jianguo Road and then walk east along Qingqin Street. The total walking distance is about 0.6 miles. After that, you can see several airport shuttle buses. The ticket price is CNY 20 per adult.

Can i get this shuttle bus from the railway station? How frequent does it arrive? May i know how long will it take from the train station to the airport?

In the railway station, you can take the airport shuttle bus Pinghai Road Line. The bus runs at an interval of 30 mins. The bus fare is CNY20 and the bus ride is about 1.5 hours.

Considering you are taking a train from Beijing, you must take the high-speed train, right? If so, you will arrive at HZ East Railway Station. And in that station, you cna also take the shuttle bus Wulinmen Line to the airport. The bus runs at an interval of 15 - 30 mins. The bus fare is the same, CNY20, and the travel time is about 1 hours.

I would like to know if I am suppose to take train from Beijing Station to Hangzhou station or Hangzhou East? I need to go to the Hangzhou International Airport.

You are advised to take high-speed train at Beijing South Railway Station, and get off at Hang zhou East Railway Station. The schedules are from 07:25 to 16:47. It takes around 5 to 6 hours on the way.
After arriving, you can find airport shuttle buses easily. The operating hours are from 05:00 to 20:30.