Dear emptynesters38,
It sounds like you're seasoned tourists so you actually might like Napoli. We head to Napoli every Monday for work in the jewelry sector and ironically we *never* wear jewelry or nice watches and I don't even carry a purse (we use a sling bag which goes around the shoulder and would recommend a money bag around the waist or shoulder/neck). Too many times we've read the papers about tourists getting roughed up for their watches or for their purses.
Also, someone posted here on the one of the forums that Napoletano major hotels are offering free watches for its tourists guests so that the guests can leave their watches in the hotel safe and use a cheapy watch during their stay. We were watching NTV, Napoli TV, a couple of weeks ago and a news story mentioned the same fact.
I really like Napoli. I agree that it is a bit worn and unkept but at the same time it's full of vibrancy and the bustle of a big city. there's a saying around here that the Napolitani are always happy and have suntanned faces as Napoli is also called the City of the Sun.
It's actually really much easier and tourist-friendly than it was around ten/fifteen years ago. More people tend to respect the traffic laws nowadays which means that at least some people actually will cross the street when the light is green and ditto for the cars. The usual etiquette for crossing the street is to just cross regardless if cars are coming in droves at 30+ mph. However, I do see people looking for the cross-walk (white horizontal bars marking the street) Also, more money has been poured into the tourist sector with neighborhoods improving their streets and erecting piazzas with fountains and tourist-friendly maps with "You Are Here" (moreso true in the Centro Storico district).
The Archeological Museum is actually pretty much comprised of incredibly beautiful paintings more than archeological finds IMHO. Also, if you need any help figuring out which buses to take to get wherever you need to go, it's notorious that you simply just ask anyone and they'll help you out. It's as if they operate in interrupt mode and have tons of patience to answer how to catch a bus to this or that or how to get to this place or another.
Here's a link to a bus company which does an easy-looking tour of Napoli called "City Sightseeing." We see these buses stop at all the major ANM (Napoli's Bus Agency) stops like Molo Beverello, Corso Umberto near the University (Piazza Borsa), Via Medina, etc. I believe they give out headphones which narrate what the bus passes by. Also, their motto is "Get On Get Off" so it sounds like it gives tourists quite a bit of freedom to stop if something looks interesting. The traffic is *insane* and getting around can be a bit stressful (crowded metros, buses, aboveways, etc with little if any air-conditioning in the Spring/Summer/Fall) so this bus deals with the cars and shuttles people up and down and around (we saw one go up the exclusive Via Tasso near Piazza Amedeo to take tourists to see that insanely lovely view from the ritzy Chiaia District (view of Vesuvio, Capri, city down below).
http://www.napoli.city-sightseeing.it/eng/index.htm
Also, if you like shopping and or window shopping (Gucci, Prada, Salvatore Ferragamo, etc), head over to Piazza Vittoria up the Rodeo-Drive-like Via Calabritto and then up to the right to see other nice stores. The trippy Street-Car-Named-Desire-like streetcar #1 or #4 can take you there. The streetcar is super rickety and may be completely crowded and hot but it's an adventure.
If you like, maybe you can stay in Sorrento and then do your daytrips to Pompei/Napoli using the Circumvesuviana and then a SITA bus to Amalfi/Ravello (I believe the ferries to/from Sorrento stop by the end of October?). I'm guessing that Sorrento has all these hotels fairly nearby the Circumvesuviana/SITA bus stop while in Napoli, hotels are spread out throughout the city so that to get to the Circumvesuviana, you'd need to take a taxi/bus/metro.
By the ways, the SITA buses tend to be more comfortable (padded seats, less crime, more heating/air-conditioning) than the regional trains and maybe the Circumvesuviana (more stops, less air-conditioning if any I heard).
Here are some transportation links:
SITA from/to Napoli to Sorrento/Positano/Amalfi:
http://www.sitabus.it/sitabus/campania/orarioCA2005-2006/Quadro-XIV-XV.pdf
Boats from Napoli to Sorrento/Amalfi/etc:
http://www.volaviamare.it/categoria.php?id_sottocategoria=344&id_categoria=176&lingua=
http://www.metrodelmare.com/inglese/l-linee_03.html
I'm not sure if service will continue until the end of October, but maybe you can check periodically to see if any indication of cessation of service is noted.
Circumvesuviana (click Piazza Garibaldi or Sorrento to see the timetables):
http://www.vesuviana.it/rete_ferstazinvpdf.htm
Buon Viaggio...