Correct estimator.
Generate i.i.d. uniform random variables $X,Y \sim \mathrm{Uniform}(0,1)$ and check whether $$ X^2 + Y^2 < 1. $$
The probability of this event is the area of a quarter unit circle inside the unit square: $$ \mathbb{P}(X^2 + Y^2 < 1) = \frac{\pi}{4}. $$
Let
$$
A_n = \sum_{i=1}^n X_i,
$$
where $X_i$ is the indicator of the event ${X_i^2 + Y_i^2 < 1}$.
Then
$$
\mathbb{E}[X_i] = \frac{\pi}{4},
$$
so the estimator
$$
\hat{\pi}_n = \frac{4 A_n}{n}
$$
is unbiased.
Standard deviation of the estimator.
Since $X_i$ is Bernoulli, $$ \mathrm{Var}(X_i) = \mathbb{E}[X_i^2] - (\mathbb{E}[X_i])^2 = \frac{\pi}{4} - \frac{\pi^2}{16}. $$
Thus, $$ \mathrm{Var}(A_n) = n \left( \frac{\pi}{4} - \frac{\pi^2}{16} \right). $$
For the estimator, $$ \hat{\pi}_n = \frac{4A_n}{n}, $$ we have $$ \mathrm{Var}(\hat{\pi}_n) = \frac{16}{n^2} \mathrm{Var}(A_n) = \frac{16}{n^2} \cdot n \left( \frac{\pi}{4} - \frac{\pi^2}{16} \right) = \frac{4\pi - \pi^2}{n}. $$
Therefore, the standard deviation is $$ \sqrt{\frac{4\pi - \pi^2}{n}}. $$