2^1000 is a very large value, you would have to use BigIntegers. The algorithm would be something like:
import java.math.BigInteger;
BigInteger two = new BigInteger("2");
BigInteger value = two.pow(1000);
int sum = 0;
while (value > 0) {
sum += value.remainder(new BigInteger("10"));
value = value.divide(new BigInteger("10"));
}