The quantum phase crossover in a spin ladder model with four-spin exchange is investigated. Previous studies show that the crossover cannot be detected by the singularity or finite-size analysis of ground-state observables for finite ladders. In this work, we find that the first-excited-state fidelity shows a sudden drop exactly at the crossover point, regardless of the length of the ladder. It suggests that the excited-state fidelity for finite systems is very effective in detecting the crossover in infinite systems. The underlying mechanism is understood through the analysis of re-construction of the low-lying energy levels of the systems.