Beam hopping (BH) is considered to provide a high level of flexibility to manage irregular and time-varying traffic requests in future multi-beam satellite systems. In BH optimization, adopting conventional iterative heuristics may have their own limitations in providing timely solutions, and directly using data-driven technique to approximate optimization variables may lead to constraint violation and degraded performance. In this paper, we explore a combined learning-and-optimization (LO) approach to provide an efficient, feasible, and near-optimal solution. The investigations are from the following aspects: 1) Integration of BH optimization and learning techniques; 2) Features to be learned in BH design; 3) How to address the feasibility issue incurred by machine learning. We provide numerical results and analysis to show that the learning component in LO significantly accelerates the procedure of identifying promising BH patterns, resulting in reduced computing time from seconds/minutes to milliseconds level. The identified learning feature enables high accuracy in predictions. In addition, the optimization component in LO guarantees the solution's feasibility and improves the overall performance with around 5% gap to the optimum.