You don't need to work with SharedObjects directly, Flox will do that for you.
The simplest way to use Flox would be to store everything on the "Player" object. First, you need to extend the Player class:
public class MyPlayer extends Player
public var lastX:Number;
public var lastY:Number;
Then, before (!) calling Flox.init, inform Flox about that player:
Flox.playerClass = MyPlayer;
When that's done, you can simply set properties on the player, and save it on the server:
var myPlayer:MyPlayer = Player.current as MyPlayer;
myPlayer.lastX = 100;
myPlayer.lastY = 150;
Next time you start the game, those properties will already have those values, directly after "Flox.init".
To load that player on another device / browser, you need to make a login, though. All information about that can be found in the documentation — but start with that simple test, and see if you can get it working.