$X_1,\cdots,X_n\sim \operatorname{Poisson}(\theta),\eta\triangleq P_\theta(X\leq1)\\$
Using Method of moments to estimate $$\sqrt{\frac{\eta(1-\eta)}{n}}$$ It is easy to show that $$\sqrt{\frac{\eta(1-\eta)}{n}}=\sqrt{\frac{e^{-\theta}+\theta e^{-\theta}-(e^{-\theta}+\theta e^{-\theta})^2}{n}}$$ From here, my friend tell me just use $\bar X$ to replace $\theta$. But I think that is something called "Functional invariance", which is a property of MLE only. Sadly, I can not find a moment lead directly to this function of parameter either.
your friend is right! the method of moment consist in estimating populationìs moments with empirical ones thus the first step is to express the unknown quantity to be estimated in function of population's moments.
You have that
$$\eta=e^{-\mu}(\mu+1)$$ thus $\hat{\mu}_{MM}=e^{-\overline{X}_n}(\overline{X}_n+1)$
then do the same reasoning with your $g(\eta)$
Here is another example to explain you MoM
Suppose you have the following density; $x \in (0;1)$, $\theta>0$
$$f_X(x)=\theta x^{\theta-1}$$
The first moment is $\mu=\frac{\theta}{\theta+1}$ thus
$$\theta=\frac{\mu}{1-\mu}$$
In other words, substituting $\mu$ with $\overline{X}_n$ you get your estimator
$$\hat{\theta}_{MM}=\frac{\overline{X}_n}{1-\overline{X}_n}$$