Consider the following two questions about natural numbers to see how they differ.
- Is the natural number, n > 1, prime?
- Is the natural number, n > 1, prime or composite?
The first asks whether a specific natural number is prime. Current proofs for primality of an arbitrary natural number n > 1 require representing that number such that the representation gets arbitrarily large. Since numbers can require an arbitrarily large representation and our resources are finite, we can only answer that question for a finite number of natural numbers.
The second question asks something different. It wants to know if the only possibilities for a natural number, n > 1, are that it can be either prime or composite. It is not the case that there is a third option for these natural numbers. Also it is not the case that a natural number can be both prime and composite.
Answering this question does not require representing n in such a way that the representation gets arbitrarily large. It can be answered unambiguously for all natural numbers even though that set is arbitrarily large or "infinite".
Here is a proof of the result that natural numbers (or integers) greater than 1 have only two possibilities. They can be either prime or composite. There is no third option and they can't be both for the same number. This proof was provided by dotslash: https://math.stackexchange.com/q/441906/312852
Let n be any integer that is greater than 1. Consider all pairs of positive integers r and s such that n=rs. There exist at least two such pairs, namely r=n and s=1 and r=1 and s = n. Moreover, since n=rs, all such pairs satisfy the inequalities 1≤r≤n and 1≤s≤n. If n is prime, then the two displayed pairs are the only ways to write n as rs. Otherwise, there exists a pair of positive integers r and s such that n=rs and neither r nor s equals either 1 or n. Therefore, in this case 1
Note that this did not require using an arbitrarily large representation for any natural number n explicitly. The proof used variables: n, r and s.
Also note that the question is not trivial. If I did not restrict n so that it is larger than 1, but allowed n to be greater than or equal to 1, then there are three possibilities for an arbitrary natural number. It could be prime, composite or, if it happens to be 1, a unit.
The second question is different from the first. It is not trivial. And it can be answered unambiguously for the entire set of natural numbers.
With that preliminary consider the scenario presented by the OP:
Potential infinity: Every natural number that can be proven to be a prime number or not to be a prime number belongs to a finite initial segment that is followed by infinitely many natural numbers. An infinite set is much larger than every finite set. Therefore almost all natural numbers cannot be proven to be a prime number or not to be a prime number.
The number of elements that can be proven to be prime or not, that is, the number of natural numbers for which we can answer question 1, is finite given current algorithms. It is not "potentially infinite".
Actual infinity: Every natural number that can be proven to be a prime number or not to be a prime number belongs to a finite initial segment that is followed by infinitely many natural numbers. Nevertheless all natural numbers can be proven to be a prime number or not to be a prime number.
The last sentence in the quote is true. For all natural numbers greater than 1 the proof above showed that they are either prime or composite, that is, prime or not prime. One can check that when the natural number is 1 it is a unit and so not prime. The last sentence is true even though we can only prove, given current algorithms, that only a finite number of natural numbers are prime.
The question, How is that possible?, could be answered by saying that we are asking two different things about natural numbers. On the one hand, we want to know something specific about a natural number, its primality. On the other hand, we want to tell if the natural number can be only one of two types, prime or composite, and not something else and not both at the same time.