I'd be hesitant about the Amalfi Coast in January. The reason is because it really is a summer, water type resort that closes down in the winter. You'd be hard pressed to find a place to eat. If you went to Positano, since it's fairly large, you'd find a few restaurants and bars (coffee) open for the locals, and only one on the beach, Chez Black. There are no boats going to or from there. The smaller towns are essentially closed down except for basic services for locals, like grocery stores and bars. Amalfi will have a handful of restaurants. It's not that cold, maybe around 50 during the day, but can be windy. Ravello is basically shut down. There was just one dreary old bar open when I spent a few weeks on the Amalfi Coast during the winter a few years ago. The locals would have to take a fairly long bus ride down to Amalfi to get groceries.

All along the coast, some of the best restaurants and views are in hotels, and most of them will be closed for the winter. In other words, the feeling will be sort of gloomy and shut down, and the weather can be nasty. And since the only thing to do there is be outside, bad weather leaves you hunkered down in a hotel with not much to do.

On the other hand, January is a great time to go to Italy because places that are normally tourist heavy, such as Venice, Florence, Rome, revert back to their normal state. You generally can walk right into the Vatican Museum and into the Sistine Chapel. So Florence would be a better side trip from Rome than the Amalfi Coast.

Now, you mention Naples and a day trip to Pompeii. Naples is fascinating. It doesn't really get crowded beyond it's normal population. There are many things to see and do in or outdoors, so it doesn't really matter what the weather is. Going there, with a day on a guided trip to Pompeii wouldn't be a bad idea. But Positano, Amalfi Coast, I wouldn't recommend that in January.