The solution to your question is actually with SQ (Singapore Airlines).

SQ will allow you to check in as early as 48 hours before flight time. It seems to me that the thing to do is that when you get up after a few hours of sleep at the Transit Hotel in T1, just go over to SQ check in at T2 to check in for your flight as well as to hand over your luggage.

http://www.singaporeair.com/en_UK/sg...unter-checkin/

Assuming this works for you, you now have a bit more time to wander around downtown as you don't have to deal with check in, but can just go straight to the gate when you get back to Changi Airport.