The application of consumer-grade rolling-shutter (RS) cameras in visual inertial odometry (VIO) systems deserves attention. In previous works, VIO systems with RS cameras usually adopt different dimensions of RS modeling to represent the camera motion. Although the complex camera motion expression improves accuracy, adding more variables greatly increases the computation load. Therefore, considering the limitations of camera motion modeling, this paper introduces a point feature correction strategy to improve the performance of the system in terms of both speed and accuracy. The optimized camera poses and high-frequency inertial measurement unit (IMU) are employed to correct the point features in the RS image to new pixel coordinates in the global shutter view (in the middle of the RS image readout time) by epipolar transfer. At the same time, we handle the zero velocity and abnormal cases of RS modeling to improve the robustness of the system. Furthermore, the readout time of the RS camera is self-calibrated online during the state augmentation stage and can quickly converge and stabilize to near the true value. The readout time of the RS camera is locally observable, except for two degenerate motions. Experimental results demonstrate that the proposed method outperforms the state-of-the-art methods on a public dataset in terms of accuracy and computational cost. We port our algorithm to an Android-based mobile phone and our system outputs real-time trajectories on the mobile phone. Then, we use the motion capture system to obtain the ground truth phone pose to test the performance of our algorithm.