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The proper motion of some of the faster stars is enough to get a date within about 200 years, and the positions of the planets have been searched exhaustively within that range, so I think the 13291 date must be accurate.

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Could it have been by accident? I.e. Randall just happened to draw something that happened to roughly correspond to year ~13300? How unlikely is it? Really interesting!

We can tell it is not an accident because there many independent sources of evidence, all of which point to the same era. In particular, the proper motion of stars (i.e. the shape of constellations) matches exactly in every case. This is very strong evidence because a) unlike axial procession, it isn't cyclical, and b) we can look at any star that moves sufficiently fast relative to us, and use it as independent evidence. There are dozens of such examples in the parts of the sky that are shown in the comic.

For example, in the image below the left-hand image is from the comic, while the two on the right are captured from Stellarium using two different dates, 13291 and 2013.

http://i.imgur.com/PrQgn4k.png

As a side note, it just isn't possible that Randall drew the stars by hand. They were either captured within planetarium software like Stellarium, or generated programmatically.

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