We study the nonequilibrium evolution of a one-dimensional quantum Ising chain with spatially disordered, time-dependent, transverse fields characterized by white noise correlation dynamics. We establish prethermalization in this model, showing that the quench dynamics of the on-site transverse magnetization first approaches a metastable state unaffected by noise fluctuations, and then relaxes exponentially fast toward an infinite temperature state as a result of the noise. We also consider energy transport in the model, starting from an inhomogeneous state with two domain walls which separate regions characterized by spins with opposite transverse magnetization. We observe at intermediate timescales a phenomenology akin to Anderson localization: energy remains localized within the two domain walls, until the Markovian noise destroys coherence and, accordingly, disorder-induced localization, allowing the system to relax toward the late stages of its nonequilibrium dynamics. We compare our results with the simpler case of a noisy quantum Ising chain without disorder, and we find that the prethermal plateau is a generic property of spin chains with weak noise, while the phenomenon of prethermal Anderson localization is a specific feature arising from the competition of noise and disorder in the real-time transport properties of the system.