Take the hostel from arrival to 6am, then jump into a taxi to one of the nearest hawker centre to Changi Airport, grab a bit of breakfast, walk around, then another taxi back to the airport, take a shower for about SGD$10 and go to the departure gate for your flight.
Breakfast for a Singaporean can be anything, congee or rice porride, or noodles, or roti channai/murtabak, or Delifrance, or kaya toast (coconut jam on toasted bread-yummo!)...
A cup of jet black coffeee with sugar, or just plain black. Bitter as!
You can sleep for about 4 hours, get up at 5am, go out of the airport, do an hour and a half or so of walking about, in one of the hawker centre/shopping complexes, and then another taxi to the airport.
That is what I would do, anyway.
Yes, it would entail spending a bit of $.
I am assuming you don't need a visa to enter Singapore.
I am telling you, it might be hot and humid once the sun comes up, but its worth it.
I do like Changi Airport, and yes, it can while your time away, if you are happy to sleep in the secure area, and then spend 3 hours or more, just walking around in "air cond comfort"!
But thats just the opinion of a cat.