<p>In this paper, we propose a Fourier-Jacobi rational spectral method using mapping strategies to address the telegraph equation within a two-dimensional irregular exterior domain. First, we apply a transformation based on polar coordinates to map the irregular exterior region to that outside a unit circle. We subsequently employ this transformation to reformulate the exterior problem of the telegraph equation and establish its variational formulation. Following this, we develop the Fourier-Jacobi rational spectral scheme, provide a detailed description of its numerical implementation, and analyze its convergence. Numerical experiments show that our spectral approach is straightforward to implement and achieves high-order accuracy.</p>