Greenfig, there is no end to the abuse a plant will get in the mail. It will be thrown, dropped, piled on and may end up on it's head for a greater portion of the trip, so plan ahead. I've received potted plants that were just about bare rooted from the shipping abuse. I've also received many broke tops on trees.
Fill the cup or pot to the brim and over with damp new print wrapped round and round. Then tape or use rubber bands to hold the paper in place. Make sure there is no place that the soil isn't covered because it will find it's way out. It's not a bad idea to put only the cup or pot in a plastic bag to contain the moisture, don't put the top in the bag. Another concern is the tree landing on it's head and getting smashed, breaking the top of the tree. A good way to prevent this and also protect the leaves and branches is to roll a piece of cardboard into a cylinder just a little smaller the the top of the pot or cup. Place in on the taped up news print and plastic bag then tape it to the cup or pot, don't spare the tape. Put the protected plant in the shipping box and cut the cylinder even with the end of the box. You do not want any play between the cylinder and the interior of the shipping box. You can stuff peanuts around the it to keep it from shifting or mashing. It will take a little effort but it will insure your plant will arrive safely.
Good luck,
"gene"
ps: a styrofoam cup makes a very poor shipping container and is sure to be broken during shippment.