In an ultra-deep mining cable elevator, the guideway consists of several tensioned cables from the surface to thousands of meters underground. The interaction dynamics between the cage and the flexible guideway is approximate as a spring-damping system, and the cage is subject to external disturbances. Lateral vibration suppression of the mining cable elevator is addressed in this paper, where an adaptive output-feedback boundary controller is designed for coupled hyperbolic PDEs with spatially-varying coefficients and on a time-varying domain, of which the uncontrolled boundary is coupled by an ODE subject to uncertain disturbances. The asymptotic convergence to zero of the ODE state and the boundedness of the PDE states are proved via Lyapunov analysis. The performance of the proposed controller on lateral vibration suppression of a mining cable elevator is verified in the numerical simulation.