You're mixing two things here. One is what the structure of the boundary is, e.g. armchair or zigzag. The other is what the wavefunction does at the boundary.
For your finite size cluster of carbon atoms, you have to decide what shape it has, which basically means deciding how many lattice sites you include and where you put them. This would decide whether your finite cluster has armchair or zigzag boundaries.
If you treat these boundaries as "real" system boundaries, then you have closed boundary condition. But if you want periodic boundary conditions, you have to decide on a way to map one boundary to the other, i.e., how does everything get "wrapped around".
E.g. for a rectangular lattice this is very easy, you just declare that electrons leaving the lattice to the left re-enter it on the right, basically stating that site $0$ and site $N$ are the same.
For your graphene lattice, you can do the same, you just have to be careful that the boundary of your finite sized cluster is such that this wrapping actually works. That means: For every atom on the boundary of your finite cell, there must still be a way to identify its 6 unique "nearest neighbors", some of which are "actual" neighbors, and some appear on the opposite site of the cluster.