###Consider mechanical explanations of gravitation
The feasibility of this behavior depends on the nature of gravity in your universe.
In our world, gravity was arguably first accurately described (but not explained) by classical Newtonian mechanics. In this interpretation, gravity is a mysterious force that acts upon all objects at a distance. But what's generating this force? Where does it come from?
Consider this quote from Newton himself:
That gravity should be innate, inherent, and essential to matter, so that one body may act upon another at a distance through a vacuum without the mediation of anything else, by and through which their action and force may be conveyed from one to another, is to me so great an absurdity that I believe no man who has in philosophical matters a competent faculty of thinking can ever fall into it.
(In the real world, this was eventually reconciled through Einstein's theory of General Relativity, which posits that all objects are constantly in freefall, and that mass merely warps the structure of spacetime, affecting the direction and rate of that freefall.)
The problem of how gravitational effects are generated seemingly from nowhere caused a great deal of scientific exploration down avenues that we now consider dead ends, but which at the time were serious attempts to explain gravity through solid mathematics and universally recognized mechanical processes and laws of motion.
These obsolete theories can be referred to as Mechanical or Kinetic theories of gravitation. Here are a few examples:
Screening: The universe is flooded with tiny particles or waves traveling at high speed in all directions (let's call them gravitons). An object alone in space is bombarded by these gravitons on all sides with equal force, for no net effect. But when you introduce another nearby massive object, that object screens some of these gravitons, preventing them from reaching the first object and thus upsetting the balance, so it starts to move in the direction of the screening object.
Vortex: The universe is naturally jam-packed with aether which is in constant motion. It moves in circles around massive bodies. Centrifigul force pushes fine matter to the outer edges of these aether vortices while heavy matter resists (with its greater inertia) and eventually finds itself pushed toward the center due to the pressure imbalance. The visible effect of gravity is comparable to stirring a cup of hot chocolate and watching the foam congregate in the center of the vortex thus produced.
Streams: The universe is (again) naturally jam-packed with aether, but matter acts like a sponge, soaking up the aether, and then either absorbing it (converting it into mass or energy) or transferring it into another world or dimension.
Waves and Pulsation: All bodies pulsate, sending waves through the aether. Depending on the theory, either the wavelength or the phase of pulsation determines whether two bodies will attract or repel each other. This stems from the observation that if two spheres suspended in a fluid pulsate in phase, they will attract each other, but if they pulsate out of phase, they will repel each other.
If in your world you adopt one of these mechanical explanations for gravity, then you might be able to find ways in which to explain the reversal of the effects of gravity.
In particular, if gravity is caused by all matter naturally and invisibly pulsating in phase, putting an object's natural pulsation out of phase will result in it being repelled by ordinary matter (and possibly attracted to matter pulsating at the same phase as itself).
Or if gravity is caused by streams of aether being absorbed into matter and sent into another dimension via some invisible gateway, a reversal of that flow such that streams of aether burst out of a body of matter might produce a localized repulsive effect.