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Elaborating on Richard's suggestion, I think that it is necessary (but don't know about sufficiency) for the poset $P$ to be Gorenstein* over the field $k$ used in defining the incidence algebra $A_P$. To see this, note that Koszulity of $A_P$ is equivalent to Cohen-Macaulay-ness of $P$ over $k$, by the results of Polo [23], Woodcock [29] (reference numbers from the paper that you quoted). Then Eulerian-ness is required by the numerology relating Hilbert series of a Koszul algebra to that of its quadratic dual, as in the reference [1, Lemma 2.11.1] by Beilinson, Ginsberg, and Soergel. And as Richard said, the Gorenstein* property is defined as being both Cohen-Macaulay and Eulerian.