Answer
Not regular.
Work Step by Step
$A^2=\begin{bmatrix}
0 &1 \\
1 & 0
\end{bmatrix}^2=\begin{bmatrix}
1 &0 \\
0 & 1
\end{bmatrix}=I
$
Thus $A^3=A^2A=IA=A$
Hence if $n$ is even $A^n=I$, if $n$ is odd $A^n=A$.
Thus no power of $A$ has only positive entries, so it cannot be regular.