A one-bedroom is from about $800 upwards, a two-bedroom from about $1200. Rooms are available from maybe $400. Those prices don't include electricity and water. Check out "New Resident Cayman" on Google, in case I've made a mistake.
Why are you asking these questions? If you don't have a job lined up, you won't be allowed to stay here. If you DO have a job, your employer usually has to find you a place.