You could look for service apartment. But the budget should be higher, around $20k or higher for one bedroom around 500 sq on island and Kowloon.
If you rent apartment from agents, you have to prepare to pay 2 month deposit and one month rent and agent fee(half month rent). For example, if the rent is $20k, the first month, you have to pay 70K. One year lease is required. landlord will return the two month deposit to you when the lease is over.
You will expect rent to be as following
Island > Kowloon > New Territories