In this work, we follow a quantum master equation (QME)-based time-domain formulation and derive various analytic expressions of a non-equilibrium Fermi's golden rule (NE-FGR) rate constant that reduces to the well-known FGR under proper limits under the long time and collective coupling limit. Using these NE-FGR rate expressions, we can explicitly perform simulations of the population dynamics using a non-Markovian EOM. This formulation also offers an improvement upon the existing QME-based approach in which no explicit discretization of the spectral density is performed, eliminating this source of additional numerical inaccuracy. The accuracy and efficacy of the NE-FGR approach are showcased via the population dynamics of a model polaritonic system with relatively weak system-bath coupling, by comparing it to the numerically exact HEOM results. We demonstrate that NE-FGR population dynamics are very accurate in this case, and for the case of lower polariton being initially populated, we report significant non-Markovian characteristics in the dynamics, which requires NE-FGR to capture properly.