$SL_2(\mathbb Z_3)/Z(SL_2(\mathbb Z_3)) \cong A_4$

Rather than the action on the $3$-Sylow, I think there is a more natural set on which $SL_2(\mathbb{Z}_3)/Z(SL_2(\mathbb{Z}_3))$ acts faithfully. Set $X:=\{\text{ vectorial lines in } \mathbb{F}_3\times \mathbb{F}_3\}$.

The cardinal of $X$ is easy to find, indeed, any vectorial line is given by a non-null vector and furthermore two colinear vectors give the same line hence :

$$X=\frac{\mathbb{F}_3\times \mathbb{F}_3\setminus \{0\}}{\mathbb{F}_3^*} $$

Hence $|X|=\frac{9-1}{2}=4$. It is clear that $SL_2(\mathbb{Z}_3)$ acts on $X$ furthermore a matrix $A$ will leave any line fixed if and only if it is in the center. This is easy to show (say $(e_1,e_2)$ is the canonical base of $\mathbb{F}_3\times \mathbb{F}_3$), take $A$ such a matrix then it leaves $\mathbb{F}_3e_1$, $\mathbb{F}_3e_2$ and $\mathbb{F}_3(e_1+e_2)$ invariant that is : $Ae_1=\lambda e_1$, $Ae_2=\mu e_2$ and $A(e_1+e_2)=\gamma (e_1+e_2)$, hence we have :

$$\gamma e_1+\gamma e_2=A(e_1+e_2)=Ae_1+Ae_2=\lambda e_1+\mu e_2 $$

Hence we have $\lambda=\gamma=\mu$, so that in the base $(e_1,e_2)$ the matrix of $A$ is :

$$\begin{pmatrix}\gamma& 0\\ 0& \gamma\end{pmatrix} $$

It is easy to show that this in the center of $SL_2(\mathbb{Z}_3)$. Conversly, any matrix in the center of $SL_2(\mathbb{Z}_3)$ is of this form and will fix any line. Hence we have shown that the action of $SL_2(\mathbb{Z}_3)$ on $X$ quotient through its center to give a faithfull action on $X$. In other words we have constructed an injective morphism :

$$\rho: SL_2(\mathbb{Z}_3)/Z(SL_2(\mathbb{Z}_3))\rightarrow \mathfrak{S}_X=\mathfrak{S}_4 $$

Now by cardinality calculus you have already done $SL_2(\mathbb{Z}_3)/Z(SL_2(\mathbb{Z}_3))=12$ and $\mathfrak{S}_4$ is of cardinal $24$. Hence the image of $\rho$ is a subgroup of index $2$ in $\mathfrak{S}_4$, it cannot but be the antisymmetric group $\mathfrak{A}_4$.