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Fundamental understanding of thermal transport in compounds with ultra-low thermal conductivity remains challenging, primarily due to the limitations of conventional lattice dynamics and heat transport models. In this study, we investigate the thermal transport in quasi-one-dimensional (1D) copper halide CsCu₂I₃ by employing a combination of first principles-based self-consistent phonon calculations and a dual-channel thermal transport model. Our results show that the 0-K unstable soft modes, primarily dominated by Cs and I atoms in CsCu₂I₃, can be anharmonically stabilized at ~ 75 K. Furthermore, we predict an ultra-low thermal conductivity of  along the chain axis and  along cross chain direction in CsCu₂I₃ at 300 K. Importantly, we find that an unexpected anomalous trend of increasing cross-chain thermal conductivity with increasing temperature for CsCu₂I₃, following a temperature dependence of ~T ^0.15, which is atypical for a single crystal. The peculiar temperature-dependent behavior of thermal conductivity is elucidated by the dominant wave-like tunnelling of phonons in thermal transport of CsCu₂I₃ along cross-chain direction. In contrast, particle-like phonon propagation primarily contributes to the chain-axis thermal conductivity across the entire temperature range of 300-700 K. The sharp difference in the dominant thermal transport channels between the two crystallographic directions can be attributed to the unique chain-like quasi-1D structure of CsCu₂I₃. Our study not only illustrates the microscopic mechanisms of thermal transport in CsCu₂I₃ but also paves the way for searching for and designing materials with ultra-low thermal conductivity.