I would not recommend to leave the airport in the wee-hours of the night. Even Singapore is pretty much sleeping then and not as bustling as some people may think.

You wont reach the recommended places much before 1.30am and you would be quite dispointed in what you find there. It mainly leftovers from the party night taking supper or patrons of the neighboring karaoke bars.
Except of Mustafa which might be a good experience to see, but if you have nothing to buy....

Stay in the transit hotel and take some rest. Thats what i would do.