I am not totally sure what you mean by "does it go twice on the stack" but the trigger happens as soon as it enters the battlefield and there is only one trigger. Getting rid of it either before or after the trigger resolves via Fling will not prevent the trigger from resolving or from its effect taking place. As long as the trigger resolves, you will still skip two turns in that case.
If you want to Stifle it, you need to do so while the trigger is on the stack immediately after Eater of Days enters the battlefield. If you do, you will not skip two turns (skipping both turns is part of one trigger).
Well what was misleading to me was the way final fortune works where i read it triggers once to give you an extra turn and another at the end of the extra turn where you lose the game ....unless stifle
No, Final Fortune is somewhat different. It has its effect when it resolves (take an extra turn). This is not a trigger; it is the spell resolving. Then, it also sets up a delayed trigger that will occur at the end of the extra turn. That is a trigger and that can be stifled. When you get to the end of the extra turn, the delayed trigger will "fire" and be put onto the stack. You can Stifle it and, if you don't, it will resolve and cause you to lose the game.