This paper concerns modeling the coupled within-host population dynamics of virus and CTL (Cytotoxic T Lymphocyte) immune response. There is substantial evidence that the CTL immune response plays a crucial role in controlling HIV in infected patients. Recent experimental studies have demonstrated that certain CTL variants can recognize HIV infected cells early in the infected cell lifecycle before viral production, while other CTLs only detect viral proteins (epitopes) presented on the surface of infected cells after viral production. The kinetics of epitope presentation and immune recognition can impact the efficacy of the immune response. We extend previous virus models to include cell infection-age structure in the infected cell compartment and immune response killing/activation rates of a PDE-ODE system. We characterize solutions to our system utilizing semigroup theory, determine equilibria and reproduction numbers, and prove stability and persistence results. Numerical simulations show that ' early immune recognition' precipitates both enhanced viral control and sustained oscillations via a Hopf bifurcation. In addition to inducing oscillatory dynamics, considering immune process rates to be functions of cell infection-age can also lead to coexistence of multiple distinct immune effector populations.