Berth allocation and scheduling at marine container terminals (MCTs) are crucial for optimizing port operations and maintaining the efficiency of maritime logistics under the pressures of increasing global trade flows and operational complexities. One of the most concerning challenges associated with berth allocation and scheduling is its environmental impact, particularly the carbon dioxide (CO2) emissions generated during port operations. This study introduces a Customized Hyperheuristic Algorithm (CHA) to solve a mathematical optimization model formulated for the green berth allocation and scheduling problem, which integrates environmental considerations to minimize both the operational costs and the CO2 emissions during vessel waiting and handling operations. CHA employs a dynamic selection mechanism that adaptively chooses heuristic operators based on their ongoing performance, enhancing the traditional framework of typical heuristic and metaheuristic algorithms. Detailed computational experiments compared CHA with established optimization techniques, such as CPLEX, ant colony optimization (ACO), simulated annealing (SA), tabu search (TS), and evolutionary algorithm (EA), which were all customized for the specific settings of the problem. It was found that CHA was able to match the objective values returned by CPLEX for small instances. Moreover, CHA outperformed ACO, SA, TS, and traditional EA by up to 8.22%, 6.96%, 9.32%, and 4.34% for large-scale instances in terms of the best values of the objective function recorded over five consecutive replications. These objective function improvements also resulted in lower emissions. Additional experiments presented some insights into port management. This study clearly demonstrated the applicability and effectiveness of CHA for sustainable and environment-friendly berth planning in real-world marine terminal operations.