Prey-predator interactions have long been central to mathematical biology studies because they are fundamental components of ecological networks. Fear effect is a phenomenon used for indirect effects of predators on prey population leading to behavioral changes. These fear-induced behaviors can cause stress-related physiological alterations, adversely impacting the body condition of the prey. This, in turn, can affect both their reproduction rates and overall survival. In this study, we revisit the inclusion of the fear effect in a prey-predator system, paying particular attention to the presence of non-fertile prey populations and how this dynamic influences the overall interaction. In essence, since the fear effect has been modeled as a growth-suppressing factor, our initial focus is to emphasize that its overall impact is diminished because non-fertile prey do not contribute to population growth. This consideration leads us to explore the model using different functional response functions to better understand the dynamics. We also choose to employ a general memory effect into the system using non-local properties of Caputo fractional derivative. The stability of the equilibrium points is analyzed. Additionally, we investigate the presence of Hopf bifurcation around the positive equilibrium point, which can indicate the onset of oscillatory behavior in the population. The theoretical findings are supported by numerical examples, providing concrete illustrations of the concepts discussed. These results are then interpreted in a biological context.