This paper is concerned with a predator-prey model with stage structure for the predator, with a cross-diffusion term modeling the effect that mature predators move toward the direction of gradient of prey. It is first shown that the corresponding Neumann initial-boundary value problem in an $n$-dimensional bounded smooth domain possesses a unique global classical solution which is uniformly-in-time bounded for the weak cross-diffusion. It is further shown that, in the presence of cross-diffusion, the models admit threshold-type dynamics in terms of the cross-diffusion coefficient; that is, the homogenous steady state keeps stability for weak attractive prey-taxis, while the stationary patterns will occur for strong attractive prey-taxis. This implies that such cross diffusion does contribute to the rich dynamics of predator-prey model with stage structure for predators.