Nice problem!

Just bumping this question

If it has $k$ columns then the sum of the entries of each row is at least $\frac{k(k+1)}{2}$, so the sum of the entries of the array is at least $4\frac{k(k+1)}{2}=2k(k+1)$. Since the sum of the entries of each column is $20$, the sum of the entries of the array is $20k$. Hence $20k\geq 2k(k+1)$, so $10\geq k+1$ and therefore $9\geq k$. The array$$\begin{array}{|c|c|c|c|c|c|c|c|c|}\hline 1&2&3&4&5&6&7&8&9\\\hline 1&2&3&4&5&6&7&8&9\\\hline 9&8&7&6&5&4&3&2&1\\\hline 9&8&7&6&5&4&3&2&1\\\hline \end{array}$$has $9$ columns and satisfies the condition. Hence the maximum is possible number of columns is $9$.