Your best bet is to stay in student housing on campus. If that is not possible, look into hostels or off-campus student housing.
There are apartments in Singapore (commonly known as HDBs), but most landlords will not allow short-term leases. It is possible to find one that will, but usually the rental will be much more than the aforementioned options.
In general, the more willing you are to spend time commuting, the cheaper your rent will be. Housing near campuses tends to be much more expensive than housing out in random suburbs.
To search for places, try a couple of these sites:
www.hostels.com
www.hostelbookers.com
www.rentinsingapore.com
www.propertyguru.com
www.singapore.gumtree.sg
even www.indiaboard.com
There are other sites, but I cannot remember them all. These would be a good start though.
By the way, a lot of people don't like hostels, but I know an Indian who lived in one for a year while doing his MBA in Singapore. I think for short-term, a hostel might be a good way to go. Food on campus is highly subsidized, so you don't necessarily need to have full cooking amenities in your rental. In fact, many landlords do not even allow cooking. If you plan to cook, make sure you confirm your ability to do so in your rental unit.