Operator learning for time-dependent partial differential equations (PDEs) has seen rapid progress, enabling efficient modeling of complex spatiotemporal dynamics. However, most existing approaches use fixed time step sizes during rollout, limiting their ability to adapt to varying temporal complexity and leading to error accumulation. We introduce COAST (Causal Operator with Adaptive Solver Transformer), a novel operator learning framework that integrates causal attention with adaptive time stepping. COAST jointly predicts the next step size and the corresponding future system state. The learned step sizes dynamically adapt both across and within trajectories—assigning smaller step sizes to regions with rapidly changing dynamics and larger steps to smoother transitions. We evaluate the COAST across a range of dynamical systems, which consistently outperforms state-of-the-art methods in both accuracy and efficiency, demonstrating the potential of causal transformers for adaptive operator learning in time-dependent systems.