Monica,
In taking a day trip from a cruise ship on your own, one of your biggest concerns will be getting back before the ship weighs anchor. Therefore your transportation links will have to be planned very precisely (and hope there isn't a rail strike that day)!
With such a very short time frame, I wouldn't recommend hiking any of the trails. Only two segments of the Sentiero Azzurro are currently open (not likely to change in the near future), you'll need to pay for a Park Pass to hike those and the two segments that are open are the most challenging of the four. You won't be strolling along the famous Via dell' Amore (Riomaggiore - Manarola) or the stretch from Manarola to Corniglia as they're both closed indefinitely. There are many other trails that extend higher in the hills but they're more difficult and will take longer, so not ideal for a day trip of only a few short hours.
One other point to consider is that the towns (especially Vernazza) will probably be dreadfully crowded with those passengers that took one of the organized excursions from the ship. There may also be one or more ships in port that day, which will exacerbate the problem. Therefore getting about on the local trains may take a bit longer and the cars will be crowded (I've experienced that first-hand). The Cinque Terre Express runs about every half hour, so that may help to alleviate the crowding.
The above comments were the cautionary note. The reality is that you should easily be able to visit all five towns in one day and at least have a quick look by using the local trains. Since you'll be docking in Livorno (which is south of the C.T.), one possibility would be to start in Riomaggiore and work north, using either the trains or the Ferry (good weather only, Ferry doesn't stop in Corniglia).
If you want to see Corniglia, it would be best to use the inexpensive Shuttle up the hill from the station, rather than walking up hundreds of steps (your choice of course). The few Euro you'll spend on the Shuttle will be worth it!
Try to allow a bit more time for Monterosso as it's larger and has both a "new town" and an "old town", so will take longer to explore. If you arrive in Monterosso in the early afternoon, you could do a bit of exploring and then have an early dinner / late lunch (15:00?) at Cantina di Miky or one of the other restaurants near the station, before returning to the ship.
You could also start from the other end. Check the train schedules on Trenitalia to see which option works best.
I highly doubt that the Cinque Terrre Treno card will be cost effective in your situation, especially if you won't be hiking any of the trails. Tickets are cheap so just buy P-P as you go.