I haven’t done Rome but Pompeii is gorgeous. I highly recommend it too (and the weather in October will be gorgeous).
£2000 is definitely doable, it depends on how much luxury you want or not. I’ve just priced it for Rome during October half term and flights are £625, hand luggage only which for a week will suffice. Berlin/Prague for example is only £300. This is from MAN, Liverpool and others might have cheaper options.
You can do cheap Airbnb and supermarket shops with one meal out every couple of days, or you might want breakfast included at the hotel and eating out every day. Free sights and cheap attractions, with 2-3 important splash outs would fit into that budget nicely, going into 5 museums each day at a charge of €10 pp, not doable.
Keep in mind that Italy, Austria, Germany have €. They’ll be much more expensive than Budapest, Prague etc, with cheaper currencies.
£2000 is definitely enough, but I do holidays on the cheap! I’m happy to fly at 6am, stay in an aparthotel, and eat cheap pizza and ice cream most days.