It changes every turn until you play it. It has a special glow on it and a tooltip if you mouse over it to remind you that it is Shifter Zerus.
It changes at the start of your turn. You can only play it as Zerus the turn that you draw it. So if you draw it on turn 1 and your opponent has a 2/1 out, you can play it as a 1/1 to contest if you want to, or hang on to it for the right time.