Synchronous update schemes in p-bit annealing offer a natural route to massive parallelism, but they can also induce period-2 oscillations that degrade optimization performance. In practical solvers, such oscillations matter only if they become observable within the finite runtime of the device or simulation. However, most existing analyses are formulated in terms of asymptotic stability and therefore do not directly address when oscillatory modes appear during finite-duration annealing. Here we develop a finite-time observability framework for synchronous tick-random p-bit dynamics. Starting from a linearized mean-field description, we derive a graph-dependent criterion that predicts whether unstable modes amplify sufficiently within a finite observation window to produce visible signatures in quantities such as the one-step autocorrelation and the energy trace. This shifts the analysis from asymptotic instability to practical detectability and yields a principled estimate of the reduction in synchrony required to suppress oscillations. We validate the framework on benchmark Max-Cut graph instances and illustrative graph families. The predicted thresholds capture both the graph dependence of oscillation onset and the finite-time conditions under which oscillations become observable. These results provide a graph-aware basis for selecting the update probability in synchronous p-bit annealers without relying on exhaustive instance-by-instance parameter sweeps.