The bulk and surface dynamics of Cu50Zr50 metallic glass were studied using classical molecular dynamics (MD) simulations. As the alloy undergoes cooling, it passes through liquid, supercooled, and glassy states. While bulk dynamics showed a marked slowing down prior to glass formation, with increasing activation energy, the slowdown in surface dynamics was relatively subtle. The surface exhibited a lower glass transition temperature than the bulk, and the dynamics preceding the transition were accurately described by a temperature-independent activation energy. Surface dynamics were much faster than bulk at a given temperature in the supercooled state, but surface and bulk dynamics were found to be very similar when compared at their respective glass transition temperatures. The manifestation of dynamical heterogeneity, as characterized by the non-Gaussian parameter and breakdown of the Stokes-Einstein equation, was also similar between bulk and surface for temperatures scaled by their respective glass transition temperatures. Individual atom motion was dominated by a cage and jump mechanism in the glassy state for both the bulk and surface. We utilize this cage and jump mechanisms to separate the activation energy for diffusion into two parts: (i) cage-breaking barrier (Q1), associated with the rearrangement of neighboring atoms to free up space and (ii) the subsequent jump barrier (Q2). It was observed that Q1 dominates Q2 for both bulk and surface diffusion, and the difference in activation energies for bulk and surface diffusion mainly arose from the differences in cage-breaking barrier Q1.