There are multiple possible cell structures on K(Z/n,1).

One is generic. For any finite group G there is a model for BG that has (|G|-1)^k new simplices in each nonzero degree k. This is the standard simplicial bar construction of K(G,1). This gives you that BG has Euler characteristic 1/|G|, if you like.

One is more specific. There is another cell structure on K(Z/n,1), viewing it as a union of generalized lens spaces, that has exactly one cell in each nonzero degree. This is a topological avatar of the "simple" resolution of Z by free Z[Z/n]-modules. Obviously this doesn't give you the Euler characteristic argument you're seeking - one needs to keep track of more intricate information about the cell attachments in order to extract something.

There are multiple possible cell structures on K(Z/n,1).

One is generic. For any finite group G there is a model for BG that has (|G|-1)^k new simplices in each nonzero degree k. This is the standard simplicial bar construction of K(G,1). This gives you that BG has Euler characteristic 1/|G|, if you like.

One is more specific. There is another cell structure on K(Z/n,1), viewing it as a union of generalized lens spaces, that has exactly one cell in each degree. This is a topological avatar of the "simple" resolution of Z by free Z[Z/n]-modules. Obviously this doesn't give you the Euler characteristic argument you're seeking - one needs to keep track of more intricate information about the cell attachments in order to extract something.