Most aerial manipulators use serial rigid-link designs, which results in large forces when initiating contacts during manipulation and could cause flight stability difficulty. This limitation could potentially be improved by the compliance of continuum manipulators. To achieve this goal, we present the novel design of a compact, lightweight, and modular cable-driven continuum manipulator for aerial drones. We then derive a complete modeling framework for its kinematics, statics, and stiffness (compliance). The modeling framework can guide the control and design problems to integrate the manipulator to aerial drones. In addition, thanks to the derived stiffness (compliance) matrix, and using a low-cost IMU sensor to capture deformation angles, we present a simple method to estimate manipulation force at the tip of the manipulator. We report preliminary experimental validations of the hardware prototype, providing insights on its manipulation feasibility. We also report preliminary results of the IMU-based force estimation method.