Quantum phase transitions are ubiquitously present in the ground and excited states of quantum many-body systems and are closely related to non-equilibrium dynamical phase transitions, but are difficult to identify. In systems of spin-1 Bose-Einstein condensates, dynamic phase transitions corresponding to equilibrium phase transitions at the ground and top excited states have been investigated, but those occurring at intermediate excited states have been untouched by previous experiments. stay. Here, we elucidate that both ground-state and excited-state quantum phase transitions in spinor condensates can be diagnosed by dynamic phase transitions. The relationship between the equilibrium phase transition and non-equilibrium behavior of the system is revealed by the quantum Fisher information. We also show that near-critical point parameter estimation beyond the standard quantum limit can be implemented. This work not only advances the investigation of quantum phase transitions of excited states via a scheme that is readily applicable to a wide class of few-mode quantum systems, but also provides a new perspective on the relationship between quantum criticality and quantum-enhanced sensing.