We study the homogenization for a class of non-symmetric stable-like Feller processes, and demonstrate its usage in the numerical approximation of first exit time. The jump intensity involves periodic and aperiodic constituents, as well as oscillating and non-oscillating constituents. This means that the noise can come both from the underlying periodic medium and from external environments, and is allowed to have different scales. It turns out that the stable-like process converges in distribution, as the scaling parameter goes to zero, to a Lévy process. As special cases of our result, some homogenization problems studied in previous works can be recovered. Moreover, we present some numerical experiments to illustrate our result by demonstrating that the first exit time of the original process can be approximated by that of the limit process.