Akrylis

While I have had luck with 895 doing this, personally it'd be kind of a waste to go to CP if you arrive at 7AM+. If anything, if you don't have checked luggage and would have spent the cash anyway look into something like Aerotel above D41. It's still air-side but generally you can get a room for $70-90 depending if you want the private bathroom for 6 hours and breakfast, they have a pool and a small gym and you can always add more time on if you want. You can go to their website and book ahead of time for a time slot, and they'll adjust it depending on when you check in; this at least ensures they'll hold the room for you even though I've seen walk-ups before. Customs will generally comment about leaving far later than your flight and/or mention that your checked baggage would be at lost and found.

Also in regards to Marriott elites, the last time I was at the JW they mentioned that despite the early time, if you call them a day or so in advance they will try to work with you if they have a room available that early.

In regards to the airport tour I cannot say as I've only seen it mentioned for long stopovers. However you could always look into getting one of the Singapore tourist passes and just use the MRT and public transit as the pass gives you unlimited rides for 10 SGD / day up to three days and they'll give you cash back if you turn in the card on your way out. You can get one in the MRT station downstairs I think on the T2 side and pretty much take it wherever you want to go.