Let's imagine a 51st state is accepted into the United States (not necessarily Puerto Rico). Would someone who was born in that state, before it joins the US, be considered a natural-born citizen? What if that person was already a naturalized citizen of the US before the state is accepted?
The U.S. Constitution uses but does not define the phrase "natural born citizen".
Historically, there were explicit rulings regarding persons who were born on a territory prior to it became an U.S. state.
We don't know the future, but we can safely assume that for future 51st state, there will be similar rulings.
United States Code, Title 8, Sections 1404 and 1405 (markup is mine):
- 8 U.S. Code § 1404 — Persons born in Alaska on or after March 30, 1867:
A person born in Alaska on or after March 30, 1867, except a noncitizen Indian, is a citizen of the United States at birth. A noncitizen Indian born in Alaska on or after March 30, 1867, and prior to June 2, 1924, is declared to be a citizen of the United States as of June 2, 1924. An Indian born in Alaska on or after June 2, 1924, is a citizen of the United States at birth.
- 8 U.S. Code § 1405 — Persons born in Hawaii:
A person born in Hawaii on or after August 12, 1898, and before April 30, 1900, is declared to be a citizen of the United States as of April 30, 1900. A person born in Hawaii on or after April 30, 1900, is a citizen of the United States at birth. A person who was a citizen of the Republic of Hawaii on August 12, 1898, is declared to be a citizen of the United States as of April 30, 1900.
Since this question is also tagged president, I suppose that the OP's concern is whether or not persons who were born on a territory before it (the territory) became an U.S. state can become the U.S. President.
The answer is yes.
Barry Goldwater was born in Arizona (1909) before it became a US State (1912). He was Rep. nominee for President in the 1964 election. His eligibility was challenged in the court. The Supreme Court made the ruling that he is eligible for presidency.
The above Wikipedia link contains a good summary on this matter.
The Citizenship Clause is the first sentence of Section 1 in the Fourteenth Amendment to the United States Constitution, which states that "All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside." As soon as the State was accepted in the Union they would be citizens but not before.
If they were already citizens, their status would remain as such.