The rise in low Earth orbit (LEO) satellite Internet services has led to increasing demand, often exceeding available data rates and compromising the quality of service. While deploying more satellites offers a short-term fix, designing higher-performance satellites with enhanced transmission capabilities provides a more sustainable solution. Achieving the necessary high capacity requires interconnecting multiple modem banks within a satellite payload. However, there is a notable gap in research on internal packet routing within extremely high-throughput satellites. To address this, we propose a real-time optimal flow allocation and priority queue scheduling method using online convex optimization-based model predictive control. We model the problem as a multi-commodity flow instance and employ an online interior-point method to solve the routing and scheduling optimization iteratively. This approach minimizes packet loss and supports real-time rerouting with low computational overhead. Our method is tested in simulation on a next-generation extremely high-throughput satellite model, demonstrating its effectiveness compared to a reference batch optimization and to traditional methods.