Hi, I've spent quite a bit of time in Singapore and here is the gym I used to frequent (but only for bouldering although there is a bit of rope climbing outside the actual gym...they also have another outdoor climbing wall, not sure if it's done yet though).
Climb Asia Climbing Centre Pte Ltd
60 Tessensohn Road
c/o: Civil Service Club
(65) 6292 7701http://www.climb-asia.com
Here is a link to a list of others depending on where you livehttp://www.indoorclimbing.com/singapore.html
Traffic can be horrific - I doubt you'll be buying a car if you're moving there temporarily because it's incredibly expensive and come rush hours/rain, it's also really difficult to catch a taxi and the taxi lines can be over an hour long. No kidding. The subway and buses are pretty efficient though.
For climbing outdoors, the only place that I've ever known is the Dairy Farm (which I may have photos of somewhere in my album...). It's a little oasis of a quarry in the middle of the busy city where people dirt bike, climb and just picnic. Single pitch sport lines. I found a blog of information on it here:http://www.juniortan.com/dairyfarm/beta.html
The hop over to Malaysia is worth it too to climb at Batu Caves (I have some photos of that for sure in my album too). The ride is about two hours I think and they have more of a variation there. I didn't have partners at the time and it was easy to head there on a weekend and share a rope with the friendly folks there.
The other big outdoors sports are water....you can head to the East Coast park where there is paddling, surfing, water skiing, swimming.....the place is almost empty during the weekday and way over in the F and G section. Always crowded in the B and C where the restaurants are. You can catch rides at the McDonald's there where the taxis frequent.
The other place is Sentosa beach and I never went there but heard the water might be better. I hope it doesn't disappoint you but the water/beaches aren't the best and they are surrounded by thousands of massive ships with containers for Singapore's hungry consumption.
But make the best of it - that's what I did. If you need more info or would like to meet partners, I can assist you with that. Just drop me a PM.