I just wrapped up a similar trip. I was in J, so when I arrived at SIN, I was wide awake. My basic schedule was:
12:10 arrival on DL (Terminal 1)
1:00 - 4:00 am Dnata Lounge (with shower) via DL J (arrival); also available on Priority Pass
4:30 am Drop bags at Left Luggage (airside) in Terminal 2; grab MRT into city
5:00 am - 9:00 am Tour Botanical Gardens (before it got hot)
9:00 am - 1:00 pm Shop and lunch at Orchard
1:00 pm - 2:30 pm MRT to SIN, collect bags, MRT to hotel
3:00 pm Check into hotel and fall into a deep sleep.