This paper presents the design of a dual-polarized frequency-modulated continuous wave (FMCW) radar system with a coupler-integrated lens antenna for automotive synthetic aperture radar (SAR) applications. High-resolution radar imaging is imperative for automotive use, with dual-polarized SAR imaging providing robust support in achieving this objective. However, exploration of the system design for dual-polarized FMCW SAR systems has been limited. The paper outlines the design of the aforementioned radar system, emphasizing its suitability for automotive SAR applications. It details the requisite parameters for dual-polarized automotive SAR systems and the design considerations for the dual-polarized lens antenna. Additionally, it proposes a radar signal monitoring method employing a coupler-integrated antenna to detect the radiated radar signal. Furthermore, a dual-polarization synthesis method is introduced to enhance SAR image quality. Experimental validation of the implemented dual-polarized FMCW radar with a coupler-integrated lens antenna confirms the effectiveness of the proposed system design.