Coarse-grained measurements are generally associated with information loss and reduced ability to resolve complex quantum dynamics. We show instead that observational entropy (OE) turns finite-resolution measurement statistics into a sensitive probe of both quantum criticality and classical instability. Across localization and ergodicity-breaking transitions, derivatives of OE accurately locate critical points. Estimates from both dynamical evolution and eigenstate analyses converge to the exact finite-size values beyond a finite observational-resolution threshold, showing that critical behavior remains accessible at finite observational resolution. In the chaotic regime, OE grows linearly up to the Ehrenfest time, allowing its entropy production rate to define an experimentally accessible Lyapunov exponent. A Pretty Good Measurement correction of the Husimi phase-space distribution yields quantitative agreement with the classical Lyapunov exponent in systems with smooth and singular potentials and admits a direct ancilla-assisted experimental implementation. It is further shown that an optimal observational resolution emerges from the competition between phase-space resolution and finite measurement statistics.