DISCLAIMER: the original answer has been rendered obsolete by the S7 finale. The events described in that episode are incompatible with the original story told by the Journal of the Two Sisters, therefore rendering its contents non-canon (by word of god the show is considered the main canon source and the side material is considered canon only when it doesn't contradict the show).
For future reference, the original post can be found here.
The answer to this question was just recently given in the show, during the Season 7 finale. Technically, Star Swirl has been alive during the whole show, even if he spent most of the time trapped in limbo - a place that the show itself describes as "beyond time" (so, it is up to you to decide if that time counts as "living").
During the finale, we learn that in order to banish the "Pony of Shadow" the six pillars sacrificed themselves and ended up trapped in limbo with their foe. Again, limbo is described as a place "in-within dimension", where "time doesn't flow normally". This is coherent with what we see later - the pillar are freed from their prison, and the last thing they remembered was the binding ritual. They even had to ask how much time passed, further proving that they had no perception of the passage of time while trapped.
From the details we have, it seems that object or living creatures trapped in limbo are for all purpose "removed from the time line" until they somehow escape (no need for food, no memory, no aging).
So, we are left with a new question. When did Star Swirl and the pillar banish the Shadow Pony? It must be after he got to met Celestia and Luna (they both mention taking lesson from him in the show) but before the Tree of Harmony was discovered and the elements first used (he had a role in the creation of the tree, but based on his reaction he did never saw the grown Tree or knew that the elements even existed before the season 7 finale).
This means that he and the other pillars must have got trapped in limbo with the Shadow Pony sometime after the foundation of Equestria but before Luna banishment
(Notice: technically right now we can't even be sure that Equestria was already founded by the time the pillar were banished. Star Swirl knowledge of Celestia and Luna does not imply they were already declared princesses. Still, he seem not surprised to find out they currently rule Equestria, nor he seem to wonder what is Equestria, so for now it may be safe to assume that Equestria was already founded at the time)