From the Central Limit Theorem, you found that every interesting distribution eventually becomes approximately normal. This includes the binomial distribution with mean \(\mu = n p_0\) and variance \(\sigma^2 = n p_0 (1-p_0)\text{.}\) Hence, the z-statistic
\begin{equation*}
Z = \frac{X - n p_0}{\sqrt{n p_0(1-p_0)}} = \frac{p - p_0}{\sqrt{p_0(1-p_0)/n}}
\end{equation*}
is approximately standard normal and so probabilities on this statistic can be computed as needed using the normal distribution.