The boundary of a standard 3-simplex $\Delta^3$ is homeomorphic to the 2-sphere $S^2$. Use this to give a $\Delta$-complex structure on $S^2$.
Let us denote the standard $3$-simplex by $[1,2,3,4]$ where $1$ is the top vertex. Then the boundary of this simplex is $[234]-[134]+[124]-[123]$. How do we use this to give a $\Delta$-complex structure on the two sphere?
To produce the desired $\Delta$-complex structure on $S^2$, you need to specify the collection of maps $\sigma_\alpha : \Delta^n \to X$, where $n$ depends on $\alpha$. To specify this collection, you need two ingredients.
First, you need the $\Delta$-complex structure on the boundary of $\Delta^3$, specified by a collection of maps which I'll write as $\tau_\alpha : \Delta^n \to X$. In your answer, you have written out the four maps of this kind for $n=2$, namely $[234]$, and $[134]$, and $[124]$, and $[123]$. You also need six maps for $n=1$, namely $[12]$, $[13]$, $[14]$, $[23]$, $[24]$, $[34]$, and four maps for $n=0$ namely $[1]$, $[2]$, $[3]$, $[4]$.
Second, you need a homeomorphism between the boundary of $\Delta^3$ and $S^2$. From the statement of your post, I assume that you know such a homeomorphism exists, and finding a formula for that homeomorphim is not what you are asking about. So I'll just use $f : \partial \Delta^3 \to S^2$ to represent that homeomorphism.
Now we put these together: the $\Delta$ complex structure is defined by the compositions $$\sigma_\alpha : \Delta^n \xrightarrow{\tau_\alpha} \partial\Delta^3 \xrightarrow{f} S^2 $$ that is, $\sigma_\alpha = f \circ \tau_\alpha$. Proving that this collection of simplices satisfies all the requirements is easy: first prove it for the stated collection of simplices $\tau_\alpha$ for $\partial\Delta^3$; and then use that $f$ is a homeomorphism to prove it for the collection of simplices $\sigma_\alpha$ for $S^2$.