Cheap = MRT to Ang Mo Kio (NS16) + 138 bus, especially if you have an EZLink card. Note that 138 bus stop to the Zoo/Night Safari is now out the West door from the MRT station and turn right to the central bus stop about 100m up the road, more or less opposite the Jubilee building. Don't cross the street. There is barely some signage in the station.
http://www.streetdirectory.com:8080/...file=&heading=

Convenient + pretty cheap: For 2 or more people, you may as well take a taxi.

Note also that returning from the Night Safari, you may want to take the special cheapo bus direct to town. Ask at the Night Safari reception counter.
http://www.busplus.com.sg/serv_ns.htm

Rich