Could you clarify what you are asking? You cut out a section of the sentence which leads me to read the first question possibly not as intended. The entire sentence reads (emphasis mine):

No Person except a natural born Citizen, or a Citizen of the United States, at the time of the Adoption of this Constitution, shall be eligible to the Office of President.

That is to say, if you were a Citizen when the Constitution was adopted, which Hamilton was, you were eligible for the Presidency regardless of where you were born. The Natural Born Citizen provision only kicks in for persons who were not citizens when the Constitution was adopted, and thus Hamilton was eligible.

So were you asking what Hamilton felt about being excluded (the answer then being that he wasn't excluded), or what Hamilton, not being born in the US himself, felt about the prospect of exclusion for future generations of immigrants?