Copenhagen to Rome sounds lovely. Our cruise on the Westerdam started in Rome then called at Valetta, Dubrovnik, Messina, Naples, Livorno, Monte Carlo, Ajaccio and Barcelona so I don't know if any of those places are on your itinery. If they are, I'm sure you'll love them. We went in early November (last Med cruise of the season) which meant ports of call were nice and quiet, but the weather was a bit wet now and then, and the sea was rough occasionally, so I think September should be a much nicer time to go weatherwise.
The Westerdam is lovely. It was our first cruise so I have nothing to compare it with, but when talking to fellow passengers who were experienced cruisers, we were assured we had "started at the top". I could ramble on for ages about the ship, but if you have any specific questions about it, then I'll do my best to answer them.
I hope you have a wonderful trip. Our holiday was the best one we've ever had and I would go again tomorrow.