Neglecting friction and other forces (air resistance) and assuming all forces are orthogonal the gravity vector , consider a large dense sphere at rest on a perfectly flat plane which is plumb. Now for the sake of simplicity, let us say the inertia of the sphere is 1 kg/m2. What happens when a force of less than the sphere's inertia is applied to the sphere? And if a force greater than the sphere's inertia is applied, will it move instantly at the velocity of the applied force, or will the velocity of the applied force dictate the period during which inertia is overcome, prior to the sphere's movement?
Neglecting friction and air resistence completely, there is no such thing as "overcoming inertia." Any external force acting on an object with mass, even if it a person pushing on the planet Earth, will cause it to accelerate, though inversely proportional to its mass (so really really small). Inertia is this concept of how much a force inversely affects acceleration, which is directly proportional to mass. It does not have units of kg/m^2
When "overcoming inertia" causes something unmoving to move, you are actually overcoming static friction.
Also note that a sphere cannot roll without friction, so it would just slide. In your example, no matter what, the sphere will slide if you push on it.
You don't have to "overcome inertia". Inertia is not something to overcome.
Inertia only slows down the acceleration which a force causes. But it doesn't prevent it. It doesn't give any lower limit that must be overcome. Look at Newtons 2nd law with mass $m$ being the inertia of the object that is pushed upon:
The tiniest force would cause a tiny acceleration. With a high inertia, the acceleration would be even smaller - but not zero.
The sphere you describe will start movig at any tiny force being applied.