I've been to SIN a few times in the past but never transiting.
I'll be arriving at 12:20am on UA continuing to Bali on a separate ticket.
Either Jetstar (4:55am) or Tigerdirect (7:00am). The first is a little cheaper and I think the connection time is plenty. Doing one ticket (causes me to fly Cathy) which will be much much greater in cost and I still don't think they interline.

I recall Airside is quite nice but I assume I will need to get my bag and exit customs and Jetstar or Tigerdirect checkin won't be open for several hours so I'll be stuck landside. Is there some place I can rest? A hotel room doesn't seem worth it.

If I just carry my bag, can I avoid customs and transit directly to Jetstar (another terminal) or Tigerdirect? Tigerdirect seems to sell a "connect" service where they'll transfer your checked back, but I assume this would not include an incoming intl flight on UA.

If I recall, there is a train between terminal, but it stops running late at night.