First off newspaper is not a good idea, it is heavy and means you may have to charge over what is needed for postage.

If they will fit I would look at using the AP boxes through click & send, they still need to be double boxed though.

Certainly fill the interior of the cup but with something lighter than newspaper, you can buy rolls of foam wrap which does the job and is also much better for putting round the items than bubblewrap which can actually cause the breakage. If it is wrapped too tight the air in the bubbles can expand in the heat of the delivery van and crush the contents.

You then need to put in a box with packing peanuts or similar then that box goes inside the AP box or a satchel.