Goal-oriented communications prioritize application-driven objectives over data accuracy, enabling intelligent next-generation wireless systems. Efficient scheduling in multi-device, multi-channel systems poses significant challenges due to high-dimensional state and action spaces. We address these challenges by deriving key structural properties of the optimal solution to the goal-oriented scheduling problem, incorporating Age of Information (AoI) and channel states. Specifically, we establish the monotonicity of the optimal state value function (a measure of long-term system performance) w.r.t. channel states and prove its asymptotic convexity w.r.t. AoI states. Additionally, we derive the monotonicity of the optimal policy w.r.t. channel states, advancing the theoretical framework for optimal scheduling. Leveraging these insights, we propose the structure-guided unified dual on-off policy DRL (SUDO-DRL), a hybrid algorithm that combines the stability of on-policy training with the sample efficiency of off-policy methods. Through a novel structural property evaluation framework, SUDO-DRL enables effective and scalable training, addressing the complexities of large-scale systems. Numerical results show SUDO-DRL improves system performance by up to 45% and reduces convergence time by 40% compared to state-of-the-art methods. It also effectively handles scheduling in much larger systems, where off-policy DRL fails and on-policy benchmarks exhibit significant performance loss, demonstrating its scalability and efficacy in goal-oriented communications.