This study develops a general analytical model for describing transient drawdown distribution induced by pumping at a finite-radius well in a radial two-zone confined aquifer of finite areal extent with Robin-type condition at both inner and outer boundaries. This model is also applicable to heat conduction problems for a composite hollow cylinder on the basis of the analogy between heat flow and groundwater flow. The time-domain solution of the model is derived by the methods of Laplace transform, Bromwich integral, and residue theorem. This new solution can reduce to the solution for constant-head test (CHT) or constant-rate test (CRT) problem by specifying appropriate coefficients at the Robin inner boundary condition. The solution describing the flow rate across the wellbore due to CHT is further developed by applying Darcy's law to the new solution. In addition, steady-state solutions for both CHT and CRT are also developed based on the approximation for Bessel functions with very small argument values. Many existing solutions for transient flow in homogeneous or two-zone finite aquifers with Dirichlet or no-flow condition at the outer boundary are shown to be special cases of the present solution. Furthermore, the sensitivity analysis is also performed to investigate the behaviors of the wellbore flow due to CHT and the aquifer drawdown induced by CRT in response to the change in each of aquifer parameters.