Per the info at Pathfinder Wiki (Shackles), the Shackles was organized in 4674 AR. Beyond that, I don't know.
/cevah
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As far as I can tell, there is no canon date given in the AP itself. The basic assumption for Golarion is that whenever you start play, the current year is 4710.
On the other hand, we have this quote:
If you never ran any of those three campaigns and you're running Shattered Star as its own, though... it's been 5 years since Runelords, 4.5 years since Crimson Throne, and 4 years since Second Darkness.
Shattered Star is the first AP to officially follow any of the previous APs. Taking RotRL as starting in early 4710, that puts CotCT starting in the middle of 4710, Second Darkness starting in early 4711, and Shattered Star starting in early 4715. Considering that Skull & Shackles was the last AP released before Shattered Star, it would make sense to have it start in 4714.
As far as month and day, we have virtually nothing to go on. The Shackles seems pretty tropical, so I don't think the starting month makes much difference in terms of weather. (Even hurricane season doesn't mean much when you have a magical hurricane nearby, throwing off tropical storms year-round.) For convenience, you could just use this year's calendar, and synchronize Day One of the campaign in-game and out-of-game.
You can take the default date by looking inside the cover for publication date, AD 2000 = 4700 AR. S&S says AD 2012. That means it'll default to 4712 AR. Rise of the Runelords defaults to second half 4707 per its original publication date (kicking off 22/9/4707), Crimson Throne to the first half of 4708, and so on. Shattered Star is I believe 2012/4712 AR.
Default AR dates:
Iron Gods - 4714
Mummy's Mask - 4714
Wrath of the Righteous - 4713
Reign of Winter - 4713
Shattered Star - 4712
Skull & Shackles - 4712
Jade Regent - 4711
Carrion Crown - 4711
Serpent's Skull - 4710
Kingmaker - 4710
Council of Thieves - 4709
Legacy of Fire - 4709
Second Darkness - 4708
Curse of the Crimson Throne - 4708
Rise of the Runelords - 4707