Yes and no. Only 1.0 players who satisfied a rather arbitrary content requirement (completing the Lunar Transmitter boss fight in version 1.0) were given the special 'Legacy' opening where their character 'warps' into Eorzea and just kind of walks into their starting city (although somewhat ridiculously wearing newbie armour and a level 1 weapon!) and are eventually remembered by npcs as being a 'Warrior of Light'. All other 1.0 players are just kind of lumped in with a new adventurer by being given the same opening of them appearing on the chocobo carriage or ferry with the merchant (never mind what we've been doing for the past five years!).

In some respects, this could be considered a lazy way of writing 1.0 characters into the storyline, but given the fact the development team were really rushing to meet the release deadline, it's understandable that it was implemented in this way as it would have eaten into valuable development time to implement three different storylines. (I created a backstory for my character explaining what happened to her in the Calamity and why she returned five years later on a chocobo carriage to Gridania wearing different clothes to what she was wearing then so it doesn't really worry me that much anymore anyway ).