Flexible cable-driven robotic arms (FCRAs) offer dexterous and compliant motion. Still, the inherent properties of cables, such as resilience, hysteresis, and friction, often lead to particular difficulties in modeling and control. This paper proposes a model predictive control (MPC) method that relies exclusively on input-output data, without a physical model, to improve the control accuracy of FCRAs. First, we develop an implicit model based on input-output data and integrate it into an MPC optimization framework. Second, a data selection algorithm (DSA) is introduced to filter the data that best characterize the system, thereby reducing the solution time per step to approximately 4 ms, which is an improvement of nearly 80%. Lastly, the influence of hyperparameters on tracking error is investigated through simulation. The proposed method has been validated on a real FCRA platform, including five-point positioning accuracy tests, a five-point response tracking test, and trajectory tracking for letter drawing. The results demonstrate that the average positioning accuracy is approximately 2.070 mm. Moreover, compared to the PID method with an average tracking error of 1.418°, the proposed method achieves an average tracking error of 0.541°.