A couple budget hotels other than the Hotel 81 chain are Fragrance hotels (another chain), Golden Landmark and Fort Canning Lodge. By budget though, I mean barely less than US$100/night. Some of them may have hourly rate rooms too but the rooms should be clean and safe. Sadly the 7th Story Hotel is now a 0 Story Hotel.
Must do in Singapore? The Zoo, Night Safari, Bird Park and Aquarium are all outstanding, IMHO and definitely worth it if you are into that. If you just want more of an asian experience, just get lost walking the lanes of Arab Street, Little India, Chinatown and Gelang Serai. Unless you are extremely careless or unlucky, there are no unsafe places in Singapore to worry about.
If you really want to sweat more, walk around Changi Village then take a bumboat over to Pulau Ubin (a small island in the northeast). Rent a S$2 bike and ride around the island to see what rural Singapore was like 50 years ago. Watch out for the wild boars and there is a great place to observe tidal marine life there while walking too. Alternatively, another very cheap idea if you like the outdoors is walking around Sungei Buloh Nature Reserve.
Anyway, Hotel 81 should work for you if you don't mind that some rooms are used by the hour in some of their hotels.
