This paper discusses the first exit and Dirichlet problems of the nonisotropic tempered $α$-stable process $X_t$. The upper bounds of all moments of the first exit position $\left|X_{τ_D}\right|$ and the first exit time $τ_D$ are firstly obtained. It is found that the probability density function of $\left|X_{τ_D}\right|$ or $τ_D$ exponentially decays with the increase of $\left|X_{τ_D}\right|$ or $τ_D$, and $\mathrm{E}\left[τ_D\right]\sim \left|\mathrm{E}\left[X_{τ_D}\right]\right|$,\ $\mathrm{E}\left[τ_D\right]\sim\mathrm{E}\left[\left|X_{τ_D}-\mathrm{E}\left[X_{τ_D}\right]\right|^2\right] $. Since $\mathrmΔ^{α/2,λ}_m$ is the infinitesimal generator of the anisotropic tempered stable process, we obtain the Feynman-Kac representation of the Dirichlet problem with the operator $\mathrmΔ^{α/2,λ}_m$. Therefore, averaging the generated trajectories of the stochastic process leads to the solution of the Dirichlet problem, which is also verified by numerical experiments.