You only need to sleep for 1-3 hours on the overnight flight to AMS. Xanax (my fave) or Ambien are your friends. Take one with dessert on the flight.

I suspect that the Singapore lounge will be your best option if you elect to stay airside, although I doubt that it will be open at 6:30am. The lounges are all at the end farthest from the 2 UA gates.

At Schiphol, all 3 "terminals" are like 1 long building. There are 3 checkin areas, though.

Although UA uses the G gates (pretty much the farthest from he B gates), the walk to the other end of the airport only takes about 10 minutes. There are ATM's along the way to get your Euros.

Personally, I would spend the time walking through Amsterdam. I arrived AMS on a Sunday morning (on UA in Business) my first time (years ago) and walking through Amsterdam at that quiet time felt like it was all mine!

BTW, the 20 minute train from AMS to/from Centraal runs every 20 minutes or so, costs about €4 each way. Tickets are cheapest from the vending machines. The same day RT ticket is the best deal. Oh, most of the vending machines that take credit cards do not take US cards. Just use the coin based vending machines. A purchase of coffee at Burger King can get you change.

Customs and Immigrations is quick and efficient at AMS. Don't worry about the time factor. No visa issues for US passports.