Final Countdown
Normal Spell
Pay 2000 Life Points. After 20 turns have passed after you activate this card (counting the turn you activate this card as the 1st turn), you win the duel.

Does "Fiber Jar" reset the effect of "Final Countdown"? You win at the end phase of the 20th turn, what is your opponent uses "Poison of the Old Man" at the 20th end phase, who wins the duel? If your end phase is somehow skipped, will you win at the 20th Main Phase II or at the 21th end phase?

Nope, never. Once Final Countdown is activated, nothing resets it. It can only be negated at the time is it first paid for.

The card says AFTER 20 turns have passed; activating Poison of the Old Man does the damage before the turn is over.

You can't skip the End Phase; that's how your turn ends. However, even if it could happen, Final Countdown resolves after the turn, not during the End Phase, so it would still work.

skocktape is correct. The countdown will continue even if "Fiber Jar" has been played. just because they say Fiber jar resets the field doesn't mean that it negates the condition that has been set for the duel.

If your opponent uses "Poison of the Old Man" during the 20th turn and takes your lifepoints down to 0, then you would lose. During the countdown, at any point if one of the players decks out or loses all their lifepoints, the other player wins. This also includes a win by Destiny Board and by Exodia.