I've never heard about the oldest reservation getting priority, either.
But the bottom line is, no matter how early you get there, and what strategies you try, there's just no way to guarantee a request. First and foremost, the room you want has to be vacant, and available to be occupied. So, in your case, there has to be someone checking out of a third floor room in Fiji the day you arrive, and those newly vacant rooms have to not need to be taken out of service for maintenance, the guest before you has to not have left the room in horrifying condition that will need special cleaning, or one of a dozen other things that could happen. And none of that is in your control. If that's the case, no amount of faxing or arriving at the crack of dawn or other strategies will help.
The resort staff does their best to accommodate guest requests. If you request a third floor room in Fiji, and there's one available on the day you arrive, in all likelihood, they'll assign it to you. If you don't get one, you can always - no matter whether you're checking in online, or in person, or if you got a text with your room assignment - always, go to the front desk and ask for a room change. If there's a third floor room in Fiji open, they'll give it to you.