Connected and autonomous vehicles have the potential to minimize energy consumption by optimizing the vehicle velocity and powertrain dynamics with Vehicle-to-Everything info en route. Existing deterministic and stochastic methods created to solve the eco-driving problem generally suffer from high computational and memory requirements, which makes online implementation challenging. This work proposes a hierarchical multi-horizon optimization framework implemented via a neural network. The neural network learns a full-route value function to account for the variability in route information and is then used to approximate the terminal cost in a receding horizon optimization. Simulations over real-world routes demonstrate that the proposed approach achieves comparable performance to a stochastic optimization solution obtained via reinforcement learning, while requiring no sophisticated training paradigm and negligible on-board memory.