Since this is a thought experiment, the material is perfectly even, we're in a perfect vacuum, and the hole is negligibly small relative to the size of the sphere, so we can ignore it for charge distribution.
But is the sphere conductive or insulating?
As you might know, for objects inside a spherical shell of constant density, the gravitational attraction cancels out. If your sphere is an insulator, then its excess electrons can't move around, and their ideally even distribution would mean that the same math applies. Once the free electrons enter the sphere, they will continue moving at constant velocity until they hit the other side of the sphere.
Whereas if your sphere is a conductor, then its electrons will move around in response to the presence of other charges, away from negative ones or towards positive ones, until their mutual repulsion balances out.
I'm not prepared to do the math, but I'm pretty sure the net effect is electric force pushing outwards from the center of the sphere. The force is zero at the exact center, but it would be like trying to balance on the top of a frictionless hemisphere. The slightest perturbation sends you inexorably in motion.
Either way, it won't work as a trap for floating charged particles. They will almost certainly hit the walls.