Ionization energy of neon vs its cationic counterpart
Does this anything to do with size of the atom shrinking?


Yes, precisely; this is one way of putting it. Another way is to say that because one electron has been removed, the remaining electrons are less shielded from the nuclear charge: therefore, the effective nuclear charge increases, and the remaining electrons are harder to remove.


  Is this because cationic counterparts always have more ionization energy than their neutral counterparts because it becomes successively more difficult to remove electrons from atoms on successive removals?


Yes. This is true for every element in the Periodic Table, with no exceptions, and it doesn't matter what the electronic configuration is.

If all else were equal, then it would indeed be harder to remove an electron from a noble gas configuration than a non-noble gas configuration. However, that is a very big if! Between $\ce{Ne}$ and $\ce{Ne+}$, the electron configurations are indeed different, but all else is not equal: as discussed previously, the effective nuclear charge in $\ce{Ne+}$ is greater than that in $\ce{Ne}$.