To address the challenges of limited dexterity in traditional transnasal surgery and inadequate intraoperative guidance， a flexible transnasal surgical robot with fiber-optic navigation is developed. The design features a flexible manipulator with spherical hinges and pin-slot configuration for enhanced dexterity and resistance to torsional disturbances. The robot's forward and inverse kinematics are modeled using geometric analysis and the D-H parameter method. Embedded with two fiber-optic sensors at 90° intervals in each manipulator， a data-driven posture calibration using binocular vision and extreme learning avoids errors from traditional offline calibration and mitigates error accumulation in differential geometry posture estimation. This enhances high-precision perception of the robot's shape and position. Experiments demonstrate the manipulator's bending angle can reach 105°， with a load capacity of 0.9 N at 80°. Using fiber-optic navigation， maximum position prediction errors are 0.920 mm in free environments and 1.635 mm with obstacles， confirming the robot's effectiveness and feasibility.