Try
www.arubamatch.com and
www.4salearuba.com (go to "editions" and choose current).
There are lots of "for rent" adds.
If you go here before you move, you can look in the supermarkets where people posts "for rents".
If you need long term, you will normally pay overprice if you book via an agent or property manager.
The local non-english papers also carry a lot of "for rent" adds.
If you don't know Aruba well, make sure to talk to people who really know Aruba, before choosing area.
Few relocated people choose to live in Oranjestad, but rather outside.
A lot of the homes in the good areas North of Oranjestad have a maid/guest house in the back, and they can be anything from a small room with bathroom up to a good size apartment.
These are often rented relatively cheaply if you are trustworthy, by people who are only here part of the year and this way you can look after the main house.
Good luck and ask away if you need more info.
Carsten