No, it gets us to an arbitrary number, rather than a specific integer or other relationship. So all we know for sure is that it is a real number. Almost all real numbers are irrational - rational numbers only turn up in the real world if there is a particular reason (for example, orbital periods of planets' moons are often related by rational numbers).
Of course, it doesn't make much sense to talk about the irrationality of a measured number, since measurements are by definition limited in precision, and irrationality is a property of the pure number. It may also turn out that it is not constant - perhaps it varies gradually with the expansion of the universe or the strength of surrounding fields.
Without any explanation or definition otherwise, however, Avogadro's number is like any other arbitrary number - irrational.
That would work...
Nope.
Unfortunately, atomic masses don't work that way. A C12 atom weighs more than 12 times an H1 atom - but not an integral (or rational) amount more.
One of pi's many definitions is the ratio of a circle's circumference to its diameter. But that is completely unrelated to the definition of a "rational number", which is the ratio between two /integers/.
I agree here - it's not /quite/ a proof...