You have two choices: either stay in the airport or go into town. Your choice will be influenced by what time you actually get through immigration and customs. This is not as clearcut as it sounds because the UA flights for the past few months have been getting into SIN late every so often due to equipment problems. In that case your time on the ground is likely to be shortened.
The quick answer is to stay at the airport. There's genuinely a ton of stuff to do at Changi. Note that most of the things to do are on the airside. There are a few food places open at the airport groundside in the middle of the night but airside is just plain more interesting to wander around even though lots of things will be closed anyway. So, just check out the airport for yourself:
http://www.changiairport.com/
If you decide to go groundside even if you stay at the airport, if you can get to the taxi stand (Singapore Immigration and customs quick and efficient) by say 1:30AM probably the simplest thing to do is to go to one of the hawker centres. Change about US$150 or so, then grab a cab and go to Newton Hawker Centre. It's more expensive than other hawker centres (because all the tourists go there) but it will be open all night and some hawker will be cooking. This way you can literally say you got some local flavor. Then get another cab to go back after say an hour or so.
About going groundside: if you decide to do this note that the UA ticket counter only opens around 4AM or so (or is it 4:30? not sure). This is important because you will need your boarding pass for the SIN/NRT flight in order to get back to airside. Even if you decide to stay airside all night I'm not sure if the transfer desk is open for UA flights at midnight. Without it you won't be able to get into the lounge; you're UA *G, right? With your UA Premier Exec card and boarding pass you can access both the Kris Gold (SQ economy) and SATS (UA contract, you get a pass) lounges. So during your transit in NRT on the way to SIN, go to the UC there and try to get them to issue you your SIN/NRT boarding pass. (Reasons for going to the lounges: Kris Gold, arguably better food but YMMV, SATS: has a shower and massage chairs.) There is the Silver Kris lounge (SQ), but you would need a business or first class boarding pass to get in.
For further info (eg what food to eat), search the other threads in this forum.
BTW, what day are you arriving and leaving SIN?