This study investigates the free vibration characteristics of a partially liquid-filled thin-walled cylindrical shell subjected to impulsive excitation. Liquid storage tanks are extensively used in industrial facilities, nuclear power plants, petrochemical industries, and water distribution systems, where their dynamic behavior under external disturbances plays a critical role in ensuring structural safety and operational reliability. The interaction between the elastic shell and the contained liquid significantly alters the dynamic characteristics of the coupled system, particularly the natural frequencies and vibration modes. In the present work, a vertical circular cylindrical shell with simply supported boundary conditions at its base and a free upper edge is considered. The contained liquid is assumed to be incompressible, inviscid, and Newtonian. The Rayleigh–Ritz method combined with Lagrange's equations is employed to derive the governing equations of motion of the coupled fluid–structure system. The hydrodynamic effect of the liquid is incorporated through the concept of added mass obtained from the solution of the Laplace equation for the fluid velocity potential. Subsequently, the free vibration problem is formulated as a generalized eigenvalue problem. Parametric investigations are conducted to examine the influence of the liquid height ratio and shell geometric characteristics on the natural frequencies of the first and second vibration modes. Numerical results indicate that decreasing the liquid filling height leads to an increase in the natural frequencies of the coupled system. Furthermore, increasing the shell aspect ratio considerably modifies the vibration response. The proposed analytical formulation provides an efficient framework for evaluating the dynamic characteristics of partially filled storage tanks and can be used as a benchmark for validating numerical simulations and supporting the design of fluid-containing shell structures.