We propose a first-order fast algorithm for the weighted max-min fair (MMF) multi-group multicast beamforming problem in large-scale systems. Utilizing the optimal multicast beamforming structure obtained recently, we convert the nonconvex MMF problem into a min-max weight minimization problem and show that it is a weakly convex problem. We propose using the projected subgradient algorithm (PSA) to solve the problem directly, instead of the conventional method that requires iteratively solving its inverse problem. We show that PSA for our problem has closed-form updates and thus is computationally cheap. Furthermore, PSA converges to a near-stationary point of our problem within finite time. Simulation results show that our PSA-based algorithm offers near-optimal performance with considerably lower computational complexity than existing methods for large-scale systems.