My recollections will be dated as I haven't even considered shipping a game since I realized it was too big a hassle and too damned expensive.
IIRC, the old go-tos were Forward Air and NAVL (Michelle). Pretty much everywhere other than NAVL required you to palletize the cabinet. Forward Air required drop-off and pick-up at air terminals. Costs started around 300 dollars and went up from there. I expect they would be higher these days. Also important to consider the "last 100 feet" in your delivery options: curbside drop-off, liftgate/no liftgate.
Having said that, a basic box will likely be cheaper, but possibly more fragile (old commercial cabs are heavy, but typically very solid).