It is somewhat difficult to get in contact with other students if you are not taking courses and spend most of your time in the lab. However, this doesn't mean you won't meet Singaporeans. You often meet a lot of interesting people heading out on the town. If you really want to meet students anyway I can highly recommend U-town, which is a university town/ campus for students at NUS. U-town is located a bus stop from NUS and you can easily get there with NUS campus buses. There is a great gym, swimming pool, great study areas and restaurants located in U-town.
Singapore has a lot to offer and you can for instance go to Sentosa, which is an amazing artificial island were you can play volley boll or just hang out on the beach. Clark Quay is a street with a lot of restaurant, bars and clubs and is often crowed at weekends. Singapore also contribute with an amazing view which you can enjoy from several roof tops, such as, Marina Bay sands night club (Ku De Ta) or 1-altitude.
If you have the budget I highly recommended you to travel since it's cheep to travel to nearby countries with Tiger airline. For instance you can travel to the phillippines within 4 hours and for only 1200kr or Bali for only 500kr. I can also recommend you to complete your diving certificate if you don't have it already. Gill Divers is a PADI certified diving center located in Clark quay with amazing instructors. We spent two afternoons with the theoretical part in Singapore and went to Malaysia over a weekend for the practical part, all arranged by Gill Divers. All in all it costs 3000 SEK which is far less from what it costs otherwise