We provide a systematic study of a noncommutative extension of the classical Anzai skew-product for the cartesian product of two copies of the unit circle to the noncommutative 2-tori. In particular, some relevant ergodic properties are proved for these quantum dynamical systems, extending the corresponding ones enjoyed by the classical Anzai skew-product. As an application, for a uniquely ergodic Anzai skew-product \Phi on the noncommutative 2-torus \AA_\a, \a\in\TT, we investigate the pointwise limit, \lim_{n\to+\infty}\frac1{n}\sum_{k=0}^{n-1}\l^{-k}\Phi^k(x), for x\in\AA_\a and \lambda a point in the unit circle, and show that there exist examples for which the limit does not exist even in the weak topology.