The Anendotan Calendar is never explained, only appearing in the Timeline page. and without some form of prior knowledge it may seem random. This is it fully explained. We shall use the date of Athena, 201X to explain the system, as it is the (relatively) most important date in the story.
Athena: Day. The days go in a consistent, 8 day loop, of Apollo, Athena, Ageleos, Hestia, Zeus, Artemis, Hephaestus, Hera.
201: "Year." The Anendotan Calendar doesn't use normal years, instead it uses what I shall call Solstice Cycles, where a year begins at either the Winter or Summer solstice, and ends at the opposing solstice. This means that the calendar progresses at approximately twice the rate of the Gregorian one.
X: Week. Instead of using months, the culture of the Anendotans decided that weeks were more important to divided the year up into. The weeks simply follow the Greek Alphabet: That is, Alpha, Beta, Gamma, Delta, and so on.
As for how to calculate the current date in Anendotan? I have no idea. We know how long an Anendotan Week is but it's very hard to properly calculate what's going on, especially since this interpretation may be completely inaccurate.