Our cruise ship docks for 12 hours near Rome on July 22nd.

It would help if we knew what you are interested in. Ancient ruins, art, the Vatican, St. Peter's Basilica, the Roman Forum, food, etc.

Be aware that with only 12 hours from docking to central Rome by train and back again, your time will be extremely limited. Unless you are the first person off the boat, you may be behind a rather large crowd of people who want, like you, to get into Rome. By train, you should plan at least an hour and a half each way, plus time for purchasing tickets. This means at least 3 hours of the 12 will be spent in travel to and from the city.

Perhaps a more practical idea would be to sign up for one of the ship excursions so that the logistics and the designated sights are included for you. If you are on an official excursion put on by the cruise company, the ship will be held until your return. If you are late returning from a self-guided visit to Rome, your ship will sail without you.

What about Ostia Antica?

I agree with Kev. Ostia antica is a great option. Hadrian's Villa, Villa d'Este, and Tivoli are also good to tour. Orvieto is a fun day trip too. It's 90 minutes from Rome via train. Your cruise ship will be docking in Civitavecchia Italy 50 miles from Rome.

Have a look at www.trenitalia.com then click on English. The trip to Orvieto with a train change in Rome takes around 2.5 hours. The return trip takes around 2.5 to 3 hours.

If your cruise ship does not depart until after 7 or 8pm then the train would be a good and inexpensive option.The train station in Orvieto is located next to the funicular which takes you to the city center.

Also, you could consider renting a car at the dock in Civitavecchia. The drive is not difficult to Orvieto since it is superhighway the whole way.

https://www.orvietoviva.com/en/

If you to want to visit towns closer to the cruise port then Tolfa and Allumiere Italy are good options. They are both less then 30 minutes from Civitavecchia.

Here’s Tom’s Port Guide to Rome for do-it-yourself.
https://www.tomsportguides.com/port-guides.html#rome

Our host Rick also has a plan for this frantic, disappointing task. You might find his Rome or Cruise Ports book in the library.

Alas, Civitavecchia is not that “close” to Rome.

Edit:
https://search.ricksteves.com/?utf8=%E2%9C%93&query=%22rome+in+a+day%22&button=

https://blog.ricksteves.com/blog/it%e2%80%99s-630-in-the-morning-welcome-to-civitavecchia-%e2%80%94-the-springboard-to-rome/

https://www.ricksteves.com/europe/italy/rome-itinerary

You'll have time for 2-3 sights at best. You also need to eat at some point! :)
I suggest you choose three sights that are geographically close to each other so you can make the most of your time there.
I would bet on not more than 7 - 8 hours in the city to be on the safe side. Where you will dock is not that close to Rome and it will take nearly an hour, if not more, to drive to the city center once you get off the ship.
I might also suggest that, rather than take the bus the cruise line offers, you should hail a taxi and do this yourself. You can time how long it takes to drive into the city and use that to roughly judge how much time it will take to get back to the ship. Always build in a cushion of time going back for traffic, etc.
You may also want to consider having a taxi driver (if you choose to take a taxi from the ship maybe that driver can do this) give you a quick driving tour around the city to show you the highlights and then drop you off near the things you decide to visit and spend more time at.
Good luck!

So it goes with the ships...you can say you were there but you aren't there long enough to do much of anything but hurry in and hurry out.
Your cruise should be offering excursions to the city. Those may be your safest bet. Since they are managed by the cruise they will get you back in time to re-board. They will offer up a few of the "highlight" Rome sites, the colosseum, the Forum, perhaps Trevi... if you chose to go on your own you need to decide what you want to see...art, ruins, Vatican, the hills and then pick. Trains work very well and will save you travel time...

Take a look at cruise critic, both the Italy board and your cruise roll call.

I have found companies that are highly recommended by other cruisers(I have done this, but only in the Caribbean) you want a company with an excellent record of getting people back to the ship on time!

While more expensive, a ship excursion may be the way to go.

Have a great trip!