I have to say I think that is a little early. This is a disembarkation, not just a day in port, so it generally takes longer. You also have some flexibility with the cruise line where you can choose early disembarkation. This usually means you are the first off the ship BUT you have to carry your own bags all the way from your stateroom to to the taxi.
Be sure to research this is being offered with the cruise line first, but it is usually offered.
Then I suggest you could ask the taxi company you are working with to suggest a time. They should know from experience what time is the best for people like you. It's not their first rodeo. If they get there early do you have to pay extra?
Also - just to add to the conversation, I would expect there to be a lot of cars available to people who walk off the ship without you having to pre-arrange one, but once again some research is called for there.
It is not a bad idea to pre-arrange a car, but my gut tells me that unless you are in a hurry to get to Rome, that is would be safer and less stressful to schedule the car for a little later, say 9:00 or 9:30. You can also get the email address of the car company and tell them you may choose to email them for an earlier pickup.
Bottom line - this is all up to you, you may be wanting to get to Rome as early as possible, so opt for early self-disembarkation (where you have to carry your own bags off the ship). Or you may choose the more relaxed approach where you pick up your bags on shore, and you can tell the ship you want to be off by a certain time (9:00 am is not unreasonable). They will then issue you tags to let you off the ship right after the first self-debarkers.