First we know that the characteristic of an ID is p, where p is prime (since ID is finite) .
Second notice that the additive order of every non zero element in the integral domain is equal to characteristic p.
Take such that . Let the additive order of x be n. Then since , n > 1. Since the characteristic is p, we know that . Thus this means that n must divide p. Since the only things that divide p are 1 and p, since we said n > 1, it must be that n = p .
This means that p must divide 256 = 2^8.
Since 2 is the only prime which divides 256. So A has char 2.