In this paper, a simple and robust shock-capturing method is developed for the Flux Reconstruction (FR) framework by combining the Adaptive Mesh Refinement (AMR) technique with the positivity-preserving property. The adaptive technique avoids the use of redundant meshes in smooth regions, while the positivity-preserving property makes the solver capable of providing numerical solutions with physical meaning. The compatibility of these two significant features relies on a novel limiter designed for mesh refinements. It ensures the positivity of solutions on all newly created cells. Therefore, the proposed method is completely positivity-preserving and thus highly robust. It performs well in solving challenging problems on highly refined meshes and allows the transition of cells at different levels to be completed within a very short distance. The performance of the proposed method is examined in various numerical experiments. When solving Euler equations, the technique of Local Artificial Diffusivity (LAD) is additionally coupled to damp oscillations. More importantly, when solving Navier-Stokes equations, the proposed method requires no auxiliaries and can provide satisfying numerical solutions directly. The implementation of the method becomes rather simple.