You could try a day trip but note travel time each way is 2.5-3.5hrs, whether by train or bus. Distance-wise, similar to visiting Malacca from Singapore. Earliest arrival would be 10:30a - 11a, which leaves ~8 hours before heading back. That's time enough to look around one town but not to go from town to town, without a car.
"The Hamptons" is roughly as large as Singapore, with far less public transit. Montauk to East Hampton is as far as Changi to Orchard MRT - a beautiful ride if you're a good biker but a bit much for a novice. Choose one town, rent a bike and enjoy the afternoon at a leisurely pace.
Montauk, at the very end of the South Fork, is more relaxed surfers' hangout than tony stomping grounds of the 1%. If you're looking for the latter, stay south of the highway, between Sagaponack and Amagansett