A member on here shipped me close to 300 of them in a small flat rate box. Believe the shipping cost was just under $6. Guessing that is why you are asking about weight. For 1,000 cases, just go with the medium size flat rate box, but I think that runs somewhere around $15 in postage.
I am pretty sure you can get around 1500 .223 cases in a medium flat rate box. I have a box around the basement with 1500 cases in it (purchased it that way).