No. He did not need to because he never ran for President. He ran for Vice-President with James Garfield in the top slot.
Garfield was shot by Charles J. Guiteau on July 2, 1881, he died until September 19, at which time Arthur was sworn in as president, serving until March 4, 1885.
He did not actively campaign for nomination for a second term, he just submitted his name for consideration. The rumors regarding his lack of natural born status made Arthur the last incumbent President to submit for renomination and not win it.
Do research or folks are gonna play you like a cheap fiddle.
So, the Constitutional provisions don’t apply if you get to be President by another methodology? Is that it?
And, it seems that we’ve found another “loophole” in the Constitution, then, if the Vice President does not have to be qualified under the Constitution to be President. Now.., that’s a real big loophole. It looks like there are more problems to fix with the Constitution... LOL...