Passive random walker dynamics is introduced on a growing surface. The walker is designed to drift upward or downward and then follow specific topological features, such as hill tops or valley bottoms, of the fluctuating surface. The passive random walker can thus be used to directly explore scaling properties of otherwise somewhat hidden topological features. For example, the walker allows us to directly measure the dynamical exponent of the underlying growth dynamics. We use the Kardar-Parisi-Zhang (KPZ) -type surface growth as an example. The world lines of a set of merging passive walkers show nontrivial coalescence behaviors and display the riverlike network structures of surface ridges in space-time. In other dynamics, such as Edwards-Wilkinson growth, this does not happen. The passive random walkers in KPZ-type surface growth are closely related to the shock waves in the noiseless Burgers equation. We also briefly discuss their relations to the passive scalar dynamics in turbulence.