To sum it up.
(SPOILER)Know those Aeons? They exist, right? They can hit monsters, be hurt by monsters, etc. They exist physically in Spira. Yuna summons the Aeons, using the power of the Fayth of that particular Aeon
Likewise, Yu-Yevon uses all those Fayth on Mt. Gagazet to summon Dream Zanarkand, and all its citizen. That's how Auron can exist in Dream Zanarkand. He walks around on a huge Aeon. That's possible because Aeons actually exist physically, like I talked about earlier. In the same way, it's possible for a part of that aeon to get away from Dream Zanarkand, and walk around physically around the rest of Spira. This is what Tidus is doing.
Tidus was carried from Dream Zanarkand (an actual physical location somewhere in Spira), to Baaj Temple. He didn't travel 1000 years through time.
Hope this explains it.