I just imagine there would be no tangible evidence to prove that this person is in fact less than 35 years old unless they were able to find witnesses to say they were less than 35.
And this could be expanded to any legal age limit in the US. How could we prove who was what age prior to inventing our current social security system?
Had such a dispute arisen, the main recourse would have been to records of the person's baptism, and to local memories of a baptism ceremony. These would suffice the establish a candidate's age to within a few weeks, and the law that required a candidate to have been born in the US would make such checks relatively straightforward.
It's worth mentioning that at least one presidential candidate did run despite being aged under 35: Victoria Woodhull, the first woman to stand for the presidency (1871-2) was only 34 years old at the time, having been born in rural Ohio in 1838. She holds a somewhat unique record, then, being doubly disqualified - by her gender and her age - from taking up the office she was running for. (Woodhull did have a neat legal argument as to why women were in fact eligible to vote and hold public office, but that's another story.)
Woodhull's candidacy is more celebrated now than it was then, and the precise details of much of her campaign are hazy. She did not stand in every state and probably did not formally appear on any ballot papers; we do not even know how many write in votes she attracted because records were not kept for such minor candidates; it must have been fewer than 5,000. There were more important reasons for disputing her right to stand for election than her age. But the fact that she was under 35 was at least noted by her opponents at the time.
As a related question: Where would the burden of proof be placed?