Auron comes from the year 0 - he was one of Yuna's father's guardians along with Jecht.
Jecht somehow got to Spira (we can only guess that it happened during the storm, when Jecht went out to sea to train). Jecht became the sacrifice so Yuna's father could summon the Final Aeon. The'fayth' of the Final Aeon becomes the next Sin - which is why Sin is Jecht. Auron, after witnessing the death of his two friends, and learning of Jecht's fate - went to confront Yunalesca (the very first summoner to defeat Sin, and the only one who can create the Final Aeon) about what happened and learnt about the endless cycle. Auron attacks her, and she kills Auron.
So Auron is basically a ghost (which is why he refused to go out into the Farplane) and therefore time and space has little meaning for him - allowing him to go pretty much whereever he wants. It also, I imagine, allows him to - in a way - communicate with Jecht - who is Sin, but doesn't want to cause any of the destruction. Jecht knows that Sin attacks in the year 1000 bc, knows his wife is dead and wants Auron to save his son.
Auron goes to the past, before the tear 1000bc and takes care of Tidus after his mother's death - so he has a father like figure, and when Sin attacks - Auron does what Jecht asks - he takes Tidus to safety in the future - where Auron hopes to stop the cycle.
Of course, Tidus was never meant to exist in the future, so he becomes part of the Fayth's dreams - therefore doesn't really exist himself... We can assume that he was probably MEANT to die in Sin's first attack.
If Yuna defeated Sin in the year 0, then Yuna's father defeated Sin 10 years previously, in the year 10bc (assuming we're counting backwards to the year 0). Sin appeared in the year 1000bc - so Yuna is not 1000 years old.