Magnetic resonance imaging (MRI) has significantly improved the monitoring of tumor ablation by providing high-contrast soft tissue images. However, MRI-guided surgical procedures often involve extremely delicate and complex manipulations, requiring an MRI-compatible robotic system to enhance precision and reduce human error from jitter during prolonged high-precision tasks. This article introduces a novel MRI-compatible, hydraulically actuated needle insertion robot for minimally invasive stereotactic neurosurgery. The robot operates based on the principles of spherical chain mechanisms and communicating vessels, being driven by a linear piston and actuator. A double-layer spherical chain mechanism translates the linear motion of the piston into rotational motion. An MRI-compatible linear encoder is constructed using a grating ruler and optical fiber to ensure precise control. Model predictive control (MPC) enabled the robot to track a given trajectory accurately, allowing for precise braking pressure management and fast response times. Testing and analysis revealed that the robotic system achieves an average positional accuracy of 1.435 mm, with angle accuracies of 0.401°. Additionally, the precision and repeatability of the robot fall within the ideal range. The results demonstrate that the developed puncture robot is both materially and structurally compatible with the MRI environment, showcasing effective spatial positioning and trajectory planning capabilities.