Highly anisotropic metal-dielectric structures reveal unique dispersion properties providing new optical effects. Here we study experimentally linear optical and magneto-optical response of arrays of plasmonic gold nanorods and similar structures complemented by a thin nickel film. We show that both types of structures reveal distinct optical features expected for hyperbolic media and associated with the epsilon-near-zero (ENZ) and epsilon-near-pole (ENP) points. In the case of Ni-containing nanocomposites, we observe linear magneto-optical effects in transmission through the structure, increasing in the vicinity of these points. This observation reveals an important role of the local field enhancement in a hyperbolic medium associated with ENZ and ENP dispersion points in the appearance of magneto-optical activity of magnetic hyperbolic metamaterials.