In this paper we discuss the relationship between direct products of monounary algebras and their components, with respect to the properties of residual finiteness, strong/weak subalgebra separability, and complete separability. For each of these properties {mathcal {P}}, we give a criterion mathcal {C_P} such that a monounary algebra A has property {mathcal {P}} if and only if it satisfies mathcal {C_P}. We also show that for a direct product Atimes B of monounary algebras, Atimes B has property {mathcal {P}} if and only if one of the following is true: either both A and B have property {mathcal {P}}, or at least one of A or B are backwards-bounded, a special property which dominates direct products and which guarantees all {mathcal {P}} hold.