Recent successful growth of asymmetric transition metal dichalcogenides via accurate manipulation of different chalcogen atoms in top and bottom surfaces demonstrates exotic electronic and chemical properties in such Janus systems. Within the framework of density functional perturbation theory, anharmonic phonon properties of monolayer Janus MoSSe sheet are explored. By considering three-phonons scattering, out-of-plane flexural acoustic (ZA) mode tends to undergo a stronger phonon scattering than transverse acoustic (TA) mode and the longitudinal acoustic (LA) mode with phonon lifetime of ZA (1.0 ps) < LA (23.8 ps) < TA (25.8 ps). This is sharply different from the symmetric MoS where flexural ZA mode has the weakest anharmonicity and is least scattered. Moreover, utilizing non-equilibrium Green function method, ballistic thermal conductance at room temperature is found to be around 0.11 nWKnm, lower than that of MoS. Our work highlights intriguing phononic properties of such MoSSe Janus layers associated with asymmetric surfaces.