A challenge in all-fiber-integrated metasurface devices is to efficiently control dispersion in the limited fiber end area to build metasurfaces, therefore, the design of metasurfaces with a special structure becomes crucial to meet the demands of dispersion control. A unique phase response of circularly polarized light in catenary metasurfaces can offer new opportunities for polarization-sensitive arbitrary chromatic dispersion control. Herein, we proposed an optical achromatic metalens based on equal width catenary metasurfaces integrated on the large-mode optical fiber (LMF) end. To reduce phase distortions, the LMF is designed to generate quasi-plane waves (QPW), and then QPW converts from catenary metasurfaces to realize achromatic focusing. A notable feature of this device is its axial focal length shift as low as 0.09% across the working wavelength range from 1.33 µm to 1.55 µm, commonly used in optical fiber communication, demonstrating its excellent dispersion control capability. Furthermore, the device exhibits exceptional capabilities to break through the diffraction limit of the output field. This research has potential applications in the fields of achromatic devices, chromatic aberration correction, fiber lasers, and optical communication and modulation.