The article presents a novel control strategy for cable-driven aerial manipulators (UAMs) aimed at enhancing impedance control during contact operations in complex environments. A fractional-order nonsingular terminal sliding mode control (FONTSMC) integrated with a disturbance observer (DOB) is proposed to improve the robustness and precision of the UAM under lumped disturbances. This developed approach utilizes the flexibility of fractional calculus, the finite-time stability of nonsingular terminal sliding mode, and the real-time disturbance estimation capabilities of the DOB to ensure smooth and compliant contact interactions. The effectiveness of the proposed control strategy is validated through comprehensive simulation studies, which demonstrate significant improvements in control performance, stability, and disturbance rejection when compared to traditional methods. The results indicate that the FONTSMC-DOB framework is highly suitable for complex aerial manipulation tasks, offering both theoretical and practical insights into the design of advanced control systems for UAMs.