Here's one option; since all vertices of the cube have names (that I asigned in the now deleted answer to the question linked), uncomment the commented out \foreach loop to see the names; I also changed the perspective and the color schema that I proposed to the other answer, but you can change it back if you want to. Since you have access to the names of the vertices, you can draw the arrows as required:
\documentclass[border=5pt]{standalone}
\usepackage{tikz}
\definecolor{myred}{RGB}{183,18,52}
\definecolor{myyellow}{RGB}{254,213,1}
\definecolor{myblue}{RGB}{0,80,198}
\definecolor{mygreen}{RGB}{0,155,72}
\begin{document}
\begin{tikzpicture}[
line join=round,
y={(-0.86cm,0.36cm)},x={(1cm,0.36cm)}, z={(0cm,1cm)},
arr/.style={-latex,ultra thick,line cap=round,shorten <= 1.5pt}
]
\coordinate (A1) at (0,0,0);
\coordinate (A2) at (0,1,0);
\coordinate (A3) at (1,1,0);
\coordinate (A4) at (1,0,0);
\coordinate (B1) at (0,0,1);
\coordinate (B2) at (0,1,1);
\coordinate (B3) at (1,1,1);
\coordinate (B4) at (1,0,1);
\fill[myyellow] (A2) -- (A3) -- (B3) -- (B2) -- cycle;
\fill[mygreen] (A2) -- (A3) -- (A4) -- (A1) -- cycle;
\fill[myred](A3) -- (B3) -- (B4) -- (A4) -- cycle;
\fill[myblue] (A1) -- (A2) -- (B2) -- (B1) -- cycle;
\draw (A2) -- (A1) -- (A4);
\draw (B2) -- (B1) -- (B4) -- (B3) -- cycle;
\draw (A1) -- (B1);
\draw (A2) -- (B2);
\draw (A4) -- (B4);
\draw[thin] (A3) -- (B3);
\draw[thin] (A3) -- (A4);
%If you want to see the names of the vertices
%\foreach \Value in {1,...,4}
%{
% \node at (A\Value) {A\Value};
% \node at (B\Value) {B\Value};
%}
\path[arr]
(A1) edge (A2)
(B2) edge (A2)
(B1) edge (B2)
(B1) edge (A1)
(B4) edge (A4)
(B3) edge (A3)
(B4) edge (B3)
(A4) edge (A3);
\end{tikzpicture}
\end{document}
