There will be no problem with immigration. You just fill out your arrival card (will be handed out on board of the aircraft), proceed to immigration (usually no queues there) and you are out of the airport in no time. On return proceed to the immigration (also no queues here) and you are back. No one will aks questions (you just recieve your arrival stamp and when back departure stamp). Just be sure that you have your boarding passes with you.

I would just head to the Esplande enjoy the skyline there (you could visit the sky terrace at the Marina Bay Resort) and have a drink or food there and than head back to the airport.

Another option would be East Coast Park. At the McDonalds (every cab driver knows the location) there are a number of food and/or beverage outlets offering from pub-food to Indian cusine. A have stroll at the beach or just relax.