I believe the night surcharge for taxis is 50%. It is in effect for cabs that depart between midnight and 6:00 a.m.

If you're trying to do things on the cheap, it might be better to book the Transport Hotel for your arrival. Single rooms without baths (you use the excellent showers/toilets in the excellent gymnasium) are SGD$40 for the first six-hour block. You can stay until about 6:00 a.m., get coffee and pastry at the airport, then take the MRT into town. Hotel + transportation for less than SGD$50!

You can do something similar on the return, but getting onto the concourse without a boarding pass is difficult. The United counter doesn't open until 4:30, and the security agents won't always accept your printed itinerary. I have been successful with paper tickets, and with itineraries/receipts that are printed on United airlines cardstock.