Accomodation for 1-2 months

Visitors without residence visa are (by law) forbidden from renting residential property, even as a temporary sublet (like AirBnB and its brethren).
Apart from a friend's place (free of charge), you can only stay at hotels, hostels and serviced apartments.
A double room at (budget) hotels starts around S$80-100/night, a more reasonable one S$150/night and up. You can find them on websites like booking.com.
Hostels are cheaper and more basic, with double rooms (no bath) starting at S$50/night. You can find then e.g. in the Loney Planet guides or at hostelz.com.
Serviced apartments usually offer the luxuries of good hotels, at a somewhat lower price of S$1000-2000/week. Google is probably the best method of finding them.