Flourish of optics of hyperbolic metamaterials (HMMs) is stimulated by their exotic optical properties. Here, we demonstrate resonant changes of the group retardation and superluminal-like propagation of femtosecond laser pulses in nanorod-based HMMs in the vicinity of epsilon-near-zero spectral point responsible for the transition between topologically distinct elliptic and hyperbolic light dispersions. Resonant dynamics of ultrashort pulses appears in a unique case when their spectral components are in both dispersion regimes simultaneously. Our findings suggest HMMs as a powerful platform for future ultrafast photonics and are pivotal for growing nonlinear optics of hyperbolic media.