Solution ahead. Don't continue if you want to solve it yourself.
×Umm, order doesn't matter, so the first row may be just arranged by increasing value. Let all the other rows be the same, then shift second row by 1, third row by 3 and fourth row by 9. Your columns now contain k, k+1, k+3, k+9 with k from 2 to A in a closed cycle. Voila. The differences show difference between each line and every other line. Since all of them are different, there will be no pairs of cards with the same value. All the calculations are modulo 13 of course.
Arrangement Differences
234567890JQKA - 1 3 9
A234567890JQK 12 - 2 8
QKA234567890J 10 11 - 6
67890JQKA2345 4 5 7 -
