We thoroughly investigate the microscopic mechanisms of the thermal transport in orthorhombic \textit{o}-CsCu$_5$S$_3$ by integrating the first-principles-based self-consistent phonon calculations (SCP) with the linearized Wigner transport equation (LWTE). Our methodology takes into account contributions to phonon energy shifts and phonon scattering rates from both three- and four-phonon processes. Additionally, it incorporates the off-diagonal terms of heat flux operators to calculate the total thermal conductivity. The predicted $\kappa_\mathrm{L}$ with an extremely weak temperature dependence following $\sim T^{-0.33}$, in good agreement with experimental values along with the parallel to the Bridgman growth direction. Such nonstandard temperature dependence of $\kappa_\mathrm{L}$ can be traced back to the dual particlelike-wavelike behavior exhibited by thermal phonons. Specifically, the coexistence of the stochastic oscillation of Cs atoms and metavalent bonding among interlayer Cu-S atoms limits the particle-like phonon propagation and enhances the wave-like tunneling of phonons. Simultaneously, the electrical transport properties are determined by employing a precise momentum relaxation-time approximation (MRTA) within the framework of the linearized Boltzmann transport equation (LBTE). By properly adjusting the carrier concentration, excellent thermoelectric performance is achieved, with a maximum thermoelectric conversion efficiency of 18.4$\%$ observed at 800 K in \textit{p}-type \textit{o}-CsCu$_5$S$_3$.} Our work not only elucidates the anomalous thermal transport behavior in the copper-based chalcogenide \textit{o}-CsCu$_5$S$_3$ but also provides insights for manipulating its thermal and electronic properties for potential thermoelectric applications.
