During loading and unloading steps, energy is consumed when cranes lift containers, while energy is often wasted when cranes drop containers. By optimizing the scheduling of cranes, it is possible to reduce energy consumption, thereby lowering operational costs and environmental impacts. In this paper, we introduce a single-crane scheduling problem with energy savings, focusing on reusing the energy from containers that have already been lifted and reducing the total energy consumption of the entire scheduling plan. We establish a basic model considering a one-dimensional storage area and provide a systematic complexity analysis of the problem. First, we investigate the connection between our problem and the semi-Eulerization problem and propose an additive approximation algorithm. Then, we present a polynomial-time Dynamic Programming (DP) algorithm for the case of bounded energy buffer and processing lengths. Next, adopting a Hamiltonian perspective, we address the general case with arbitrary energy buffer and processing lengths. We propose an exact DP algorithm and show that the variation of the problem is polynomially solvable when it can be transformed into a path cover problem on acyclic interval digraphs. We introduce a paradigm that integrates both the Eulerian and Hamiltonian perspectives, providing a robust framework for addressing the problem.