There was no actual 'tme travel' involved at all, in any way. Bevelle defeated Zanarkand during their war, in a surprise attack. Zanarkand was reduced to being the ruins visited in Spira. Yu Yevon, ruler of Zanarkand, was distraught and used his powers to create Sin from the Fayths of Zanarkand - that cluster of crystal-shrouded bodies on Mount Gagazet. These Fayths, and the separate Fayths in the Temples of Yevon, are all keeping the 'Dream Zanarkand' alive, so that its glory never fades. An imaginary world, but based on their own city. "Zanarkand never sleeps", because it's one ongoing fantasy. But the Fayths grew tired of the endless dreaming, and Sin's unrelenting vengeance out in the real world. The Fayths wanted to break the cycle and finally set themselves free of the undying dream. Sin can exist in the dream world, and seems to go there whenever it is defeated by the Final Summoning. Jecht made it into reality in this way. But then, the Fayths saw a way to end it all, for good. They chose Tidus to make the attempt; Jecht's attack on the 'Dream Zanarkand' was intended to give Tidus the drive and motivation to beat Sin - especially once Auron told him who Sin was. The Fayth of Bahamut appears to Tidus several times, basically because he's been 'chosen' to champion their cause out in reality. Auron was able to cross into the dream world, via Sin and probably with the consent of the Fayths, too.
Once Yevon was defeated, the Fayths could finally be sent, and the Fayth Cluster could be set free of its dream, too. Remember when you first reach the summit of Gagazet, and Yuna notices that someone's summoning via the Fayths up there? That column of blue-green energy, which we see dissipating at the end, is where the 'Dream Zanarkand' originates, I think. After all, when that energy finally started to fade, Tidus' face was on one of the bubbles, which all showed someone from Zanarkand.

It's complex, but I don't think there are any genuine holes in the story. Like Radje said, Maechen's explanations are very helpful and extremely revealing.