I have the following problem.
Find the proportion of prime numbers among positive integers up to and including with 500 decimal digits.
I do not have problem with math; the problem is that I do not understand what the problem is asking due to my english. Is my interpretation correct that question is asking?
Find the proportion of prime numbers among positive integers between $1$ digits and $500$ digits (including both bounds)
Your interpretation looks fine. The exact answer is
$$\pi(10^{500})\over10^{500}-1$$
(You could also write $\pi(10^{500}-1)$ in the numerator, but you don't need to, since $10^{500}$ is clearly not prime.) To check that this is correct, consider the easier example where you want the proportion up to and including $2$ digits, which means the one- and two-digit numbers $1$ to $99$, where the exact answer is
$${\pi(10^2)\over10^2-1}={\pi(100)\over99}={25\over99}$$
If an approximate answer is desired, the Prime Number Theorem tells you
$${\pi(10^{500})\over10^{500}-1}\approx{1\over\ln(10^{500})}={1\over500\ln10}\approx.000868589$$