I don't know that they all run seven days a week, but Grimaldi Lines runs overnight ferries from Catania to Salerno. Travel time is 13 hours. Basic fare is very reasonable; I don't know what it would look like by the time you pay for a cabin, but perhaps not bad, given there are at least two of you.
Tirrenia and Grandi Navi Veloce have daytime ferries from Palermo to Naples. Running time is 10-1/2 to 11 hours.
There are additional options from Sicily to the Naples area. I used the DirectFerries.com website to check schedules. I believe it is often cheaper to buy ferry tickets well in advance, as is the case with trains.