Taxi. Shuttle is possible before midnight, but, for a first trip, the cost differential given the potential hassle isn't that great. Taxi drivers are uniformly honest.

I usually get to the airport 1.5 hours ahead. Leave enough time for security at the gate, which can be take some time. The procedures at SIN keep varying, so perhaps someone with the most recent information will reply.

As to your question about what to do, are you a Lonely Planet traveller or upscale or what? Where are you staying?

I usually recommend the Night Safari, which is fairly unique. There actually are a lot of decent reports in the specific discussion areas.

You could do worse than to review the 3 Perfect Days write-up. Shopping is just ok, but the food is great.
Rich