Magnetic microrobots can be navigated by an external magnetic field to autonomously move within living organisms with complex and unstructured environments. Potential applications include drug delivery, diagnostics, and therapeutic interventions. Existing techniques commonly impart magnetic properties to the target object,or drive the robot to contact and then manipulate the object, both probably inducing physical damage. This paper considers a non-contact formulation, where the robot spins to generate a repulsive field to push the object without physical contact. Under such a formulation, the main challenge is that the motion model between the input of the magnetic field and the output velocity of the target object is commonly unknown and difficult to analyze. To deal with it, this paper proposes a data-driven-based solution. A neural network is constructed to efficiently estimate the motion model. Then, an approximate model-based optimal control scheme is developed to push the object to track a time-varying trajectory, maintaining the non-contact with distance constraints. Furthermore, a straightforward planner is introduced to assess the adaptability of non-contact manipulation in a cluttered unstructured environment. Experimental results are presented to show the tracking and navigation performance of the proposed scheme.