Say a conductor with an initial electric potential of zero is subject to an arbitrary charge. I understand that because if this outside charge, there would be charge distribution inside the conductor, so as to make the electric field in it zero. What happens to the initial electric potential inside the conductor? Would it be greater than zero since now one side of the conductor is positively charged and another negatively?
Electromagnetism – Electric Potential Inside a Polarised Conductor
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The field is actually discontinuous at the surface: the discontinuity in the field is proportional to the surface charge density.
The statement "within the conductor and the surface" is to be understood as meaning within the conductor and a point arbitrary close to the surface but inside this surface. The situation you describe is an idealization as, in real conductors, the charge is concentrated in a small boundary around the surface; the thickness of this boundary depends inversely on the conductivity of the material, and goes to zero in the ideal case of a perfect conductor with conductivity $\sigma\to\infty$.
The "NET" Electric field inside a "conductor" is always zero as you have said.This is basically due to the presence of a lot of free electrons in the conductor.
Let us consider a conducting plate kept in a uniform electric field perpendicular to the plane of the conductor as shown.
The electric field exerts a force F=-eE on the the free electrons of the conductor pulling the electrons to the left side ( to the direction opposite to the electric field) of the conductor creating negative charge accumulation on that side. This in-turn creates a positive charge accumulation on the other side of the conductor. This accumulation starts to create an electric field opposite in direction to the existing electric field ( that is the new electric field opposes the already existing one). This accumulation increases and so do the opposing electric field. This continues until both the electric field become equal in magnitude. So the new opposing electric field completely cancels out the already existing on (only inside the conductor). So the net electric field in inside the conductor becomes 0. All this happens basically in nano seconds. So, the conductor isn't affected by any new electric field coming across it as it is cancelled out instantly.
So, Getting to your question, the excess charges creates an electric field inside the conductor.Now,the mobile electrons rearranges as you have said to create another electric field which will totally cancel out the existing electric field inside the conductor. Its the "net electric field" that will be zero inside a conductor.
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Let's be a little more precise about what we mean by a zero potential. We'll take the potential of earth to be zero, and before we bring up the charge we'll connect our conductor to earth to make its potential zero as well. Then we disconnect the conductor from earth.
Now we bring up the external charge, and as you say it will polarise the conductor. The question is whether the potential of the conductor has been changed, and the simple way to test this is to connect it to earth again and see if any charge flows between earth and the conductor. If no charge flows the potential of the conductor must be unchanged, and if charge flows the potential must have changed.
And if we tried this we would find that charge does flow between earth and the conductor as soon as we connect them. If we bring up a positive charge and connect the conductor to earth we'll find electrons flow from earth onto the conductor to give it a net negative charge. Likewise if we bring up a negative charge we'll find electrons flow off the conductor to earth giving the conductor a net positive charge. Either way bringing the external charge close to the conductor does change its potential relative to earth.
Actually calculating the change in the potential would be hard, and if would depend on the size and shape of the conductor. However our thought experiment makes it clear that the potential does change.