In nuclear decommissioning and other hazardous environments, long slender remote manipulators are crucial for performing tasks where human intervention is not possible. However, their inherent flexibility poses critical challenges such as deformation and vibration, which can degrade control precision and induce risks in rapid operations. To address such issues, this paper presents a new vibration suppression control strategy for a specialised type of flexible manipulators. A two-link model, featuring a fixed, flexible first link and a rigid second link, is considered to capture the core effects of one typical kind of long-reach remote systems, which consist of a relatively rigid manipulator supported by a long slender structure. The system dynamics are modelled using the assumed mode method, and a prescribed vibration control (PVC) framework is developed, integrating two prescribed vibration control barrier functions (PV-CBFs) and a finite-time disturbance observer to manage unmodelled uncertainties and external disturbances. This framework ensures predefined vibration performance and accurate end-point positioning by formulating a quadratic programming (QP) problem that adjusts a baseline tracking controller. The proposed method is validated through rigorous theoretical analysis and simulation tests, demonstrating better tracking performance compared to traditional methods.