We have studied the impact of finite angular velocity on the viscous properties and expansion dynamics of a rotating quark-gluon plasma (QGP) medium. In the noncentral events of relativistic heavy ion collisions, the produced QGP medium carries substantial angular momentum, resulting in a measurable angular velocity that can significantly modify its viscous properties. Thus, there exist phenomenological consequences of such rotation on the shear and bulk viscosities of the medium. Using a novel relaxation time approximation in the relativistic Boltzmann transport equation and incorporating finite angular velocity, we have calculated the shear and bulk viscous coefficients of the rotating QGP medium. Our results indicate that the introduction of rotation enhances both shear and bulk viscosities, reflecting increased momentum transport and amplified local pressure fluctuations. We have further explored the hydrodynamic evolution of the medium within the Bjorken boost-invariant expansion scenario and observed that the presence of rotation leads to a faster decrease of the energy density as compared to the nonrotating case. Hence, rapid rotation enhances the cooling rate of the QGP medium. The present analysis is mainly pertinent to the early-stage hydrodynamic evolution of the QGP, where vorticity and dissipative effects play important roles.