Advice for hotels in Naples and capri

Capri- La Minerva, a boutique hotel with water views, incredible service by owners.

What is their price range, and for how long are they staying in each area?

I would look at www.booking.com and also consider Anacapri. The hotels may be (a bit) less expensive than in Capri, which is uniformly expensive. As for sightseeing, it's a fairly compact area so if they are visiting over a few days, they will likely see everything.

Here are some ideas (I think the blue grotto is over-rated but to each his own) - I enjoyed just strolling, hiking up to Villa Jovis, checking out the botanical gardens (Giardini di Augusto), and going up to Monte Solaro the most.
https://www.planetware.com/italy/capri-and-the-blue-grotto-i-cm-cp.htm

We stayed at Hotel della Piccolo Marina in Capri, and found it well located to do various things around the Island. It was a 10 minute walk from the main Capri town, but very close to the buses and funicular. The hotel and staff were lovely (we booked a deluxe double, and got upgraded to a superior suite where we each had our own bathrooms, and a lovely balcony!) A restaurant recommended to us which we enjoyed was Ristoante il Genario, which requires reservations.

We spent our honeymoon at the Hotel Caesar Augustus in Anacapri. Now THAT is a hotel! We went back five years later and got the same room. The view is astounding.