The Add to Cart buttons can be opened and edited from the Server Behaviors panel. I just opened the page and I see them all there.
To edit the code manually you just need to find the add to cart code on the top of the page and add the new column value. To do that, find any line that starts with:
$ATC_itemPrice = floatval
and then add a line below it for the new column:
$ATC_itemShipFlat = floatval("0");// column binding
Then, two lines below that, you need to add the new column to the actual add to cart function like:
$OldTheatrePosters->AddToCart($ATC_AddIfIn, $ATC_RedirectIfIn, $ATC_itemID, $ATC_itemName, $ATC_itemDescription, $ATC_itemThumbnail, $ATC_itemWeight, $ATC_itemQuantity, $ATC_itemPrice, $ATC_itemShipFlat);
You can see I added ",$ATC_itemShipFlat" before the closing ")";
You would have to do that for each add to cart code chunk. I've gone ahead and made the change for the first one on j.php by updating lines 21 and 24.