I was at NUS as an exchange student (from Sweden) about 10 years ago. A lot of things have happened since then, e.g. housing prices have skyrocketed, and some of the advice below may be outdated, but hopefully some of the info may be of use to you. On campus residences (such as Prince George's Park, where most of the exchange students lived) filled up on a first come first serve basis, so don't make the mistake of applying right before the official deadline since it may already be full then. If you have problems finding a place to stay, then go to the various offices in charge and ask in person. My general impression of the NUS bureaucracy was that the default answer would be "no", and if they told you to get back in x number of days, then that would be too late. The solution was to show up every day and bug them enough in person (with a smile!) until they helped you. It may be a bit of a culture shock, but just have a sense of humor, don't be in a rush, and appreciate the fact that you're on an adventure and expanding your world view, frustrating as it may be. Loosing your temper will get you nowhere!! I had booked an ordinary hostel for the first few days when I arrived, so I could have my jet lag in peace and didn't have to drag my stuff around campus while settling the initial paperwork.
There are MANY exchange students in Singapore, who - sadly - ended up hanging out mostly with each other (I would probably have ended up speaking more Swedish than English in my free time had I not actively made sure to befriend some locals). It makes sense considering that you primarily meet other foreigners during the introductory information sessions (excellent place to find people to share accommodations with btw) and most people are looking for travel buddies to go on weekend trips with. I recommend seeking out the "matriculation fair" right before the beginning of the semester. For some reason it was not advertised in any of the materials I received from NUS, so ask anyone about when and where it is. The matriculation fair gives you a lot of practical information regarding student discounts on computers, sim-cards etc. and it also has representatives from most student clubs on campus, which is an excellent way to meet locals! I'm not a huge sports fan, but I decided to try out some local martial arts (pencak silat), which I sucked at, but I had a great time and made some good friends who showed me many new sides of Singapore that most of the other exchange students missed out on.
You'll get by just fine with English although the Singlish takes a while to get used to. Exceptions may be some hawker centers (including on campus) where some basic Mandarin can make things easier, but pointing at what you want works just fine. If you're interested in getting to know any of the local languages I was very pleased with the Bahasa Indonesia class (almost the same language as Malay) offered at NUS. It is a very easy language (especially compared to Mandarin and Tamil), the teachers and syllabus are great, AND you have a lot of tutorials to choose from so it is easy to fit into your schedule.
As for nightlife I primarily headed downtown for clubbing, although there was usually some activity going on in Holland Village, not too far from campus. I would strongly recommend living close to campus. You can always take a taxi to get home in the middle of the night, but trying to get to class during rush hour can be a real pain.