This investigation focuses on a class of double free boundary problems pertaining to a stage-structured predator-prey model. The model features a predator population partitioned into immature and mature stages, where immature individuals are incapable of reproduction and diffusion. We start with the local existence and uniqueness, then prove global existence and uniqueness while obtaining corresponding estimates. Furthermore, we characterize the long-time behavior of solutions and demonstrate a spreading-vanishing dichotomy. Ultimately, sufficient criteria are provided to delineate the scenarios leading to spreading versus vanishing.