This appears to be how they computed the date: https://en.wikipedia.org/wiki/K%E2%80%93Ar_dating

With just 1 sample they can't conclude much, but if they find a bunch of others, they'll have an interesting set of datapoints from the composition of the tiny air pockets in the ice.

You can read more in this open-access PNAS article[0].

[0]http://www.pnas.org/content/105/24/8232.full

Does this mean it should have 2.7 million ice core layers in the cross section?

No, the area where the ice was recovered is near the Allan Hills of the Transantarctic Mountains. This area isn't in the center of the ice sheet, where consistent accumulation patterns exist and drilling from the surface to the deepest ice would yield ~500k of these annual layers. The Allan Hills are on the fringe of the ice sheet, where the rock beneath the ice rises and due to some of the dynamic deformation mechanisms (and strong surface ablation) old ice is forced towards the surface. The researchers therefore aren't drilling from modern ice to this ancient ice, but are instead tapping directly into the old ice which is now exposed near the ice sheet surface.

Apparently not. They say the ice "is not organized into neat layers", so they have approximated the age based on potassium and argon concentration.

Though it doesn't mention what potassium and argon have to do with age. Short explanation, anyone?

Isotopes of potassium decay to argon and the amount of accumulated argon - combined with knowledge of the decay rate - allows one to use the system as a clock:

https://en.wikipedia.org/wiki/K%E2%80%93Ar_dating

that assumes you know how much it started with and what the environment was like then

The Wikipedia page covers these issues, albeit for crystalized rock instead of ice. The measurement is a ratio, so the key thing is establishing that Argon 40 is normally not present in rain or brand new ice.

It's safe to assume that these assumptions have been accounted for by the scientists developing and performing isotopic dating.