This paper focuses on an eco-driving-based hierarchical robust energy management strategy for connected and automated hybrid electric vehicles in the presence of uncertainty. The proposed control strategy includes a velocity optimizer, which evaluates the optimal vehicle velocity, and a powertrain energy manager, which evaluates the optimal power split between the engine and the battery in a hierarchical framework. The velocity optimizer accounts for regenerative braking and minimizes the total driving power and friction braking over a short control horizon. The hierarchical powertrain energy manager employs a long- and short-term strategy where it first approximately solves its problem over a long time horizon (the whole trip time in this paper) using the traffic data obtained from vehicle-to-infrastructure (V2I) connectivity. This is followed by a short-term decision maker that utilizes the velocity optimizer and long-term solution, and solves the energy management problem over a relatively short time horizon using robust model predictive control (MPC) methods to factor in any uncertainty in the velocity profile due to uncertain traffic. We solve the long-term energy management problem using pseudo-spectral optimal control method, and the short-term problem using robust tube-based MPC method. Simulation results with standard driving cycle velocity profile and real-world traffic data show the competence of our proposed approach. Our proposed co-optimization approach with long- and short-term solution results in more energy efficiency than a baseline co-optimization approach.