Mixed-dimensional partial differential equations (PDEs), characterized by coupling between operators defined on domains of varying dimensions, pose significant computational challenges due to their inherent ill-conditioning. Furthermore, the computational effort rises as the problem needs to be solved multiple times to address various instances of the low-dimensional problem, that, in applications, pertain to the description of fracture, fiber, or vascular networks configuration. In this work, we propose a novel, matrix-free preconditioning strategy that leverages operator learning to efficiently address a class of 3D-1D mixed-dimensional PDEs. The proposed preconditioner generalizes across varying shapes of the 1D manifold without retraining procedure, making it robust to changes in graph topology. This study establishes a foundation for extending machine learning-based preconditioning techniques to broader classes of coupled multi-physics systems, providing a powerful tool for overcoming complex computational challenges in scientific computing.