A simple minded Poincare duality for orbifolds?

Your proof for global quotients is correct, although the homology is more naturally the co-invariants, though these are isomorphic to the invariants by the transfer. The dual of the co-invariants is naturally the invariants of the dual. So to say that the co-invariants of two dual spaces are again dual, you need to adjust the pairing by the the order of the group, though people don't usually worry about such factors.

For the general case, a useful tool is homology manifolds. This is a local property which the underlying (or "coarse") space of an orbifold has with $\mathbb Q$ or $\mathbb R$ coefficients. The same argument you applied to the homology in the global quotient case applies to the local homology. Homology manifolds have Poincare duality. Unfortunately, the definition I linked in wikipedia is not quite correct. Your user name suggests that you might like sheaf theory. The correct definition is that a certain sheaf is constant; the incorrect definition checks only the stalks. In fact, this sheaf is the dualizing sheaf, though sheaves are overkill to proving PD for homology manifolds.

Chapter 5 of "Orbifolds and stringy topology" by Adem, Leida, Ruan have a version of Poincare duality in the groupoid setting. This is probably a very general result on this.