Venice cruise ship port Trieste

About 120 km as the crow flies. There are trains between Venice and Trieste that can be used by anyone with a ticket, including cruise ship passengers.

You will have to get a taxi from the cruise port to Trieste Centrale railway station, then a train to Venezia Santa Lucia railway station.
The port looks to be about four kms from the rail station, if I'm looking at the right spot on a map, and a taxi should take about 12 minutes.
I don't know where a taxi rank would be, near the port.
Depending on the train, the train will take from just over two hours till three and a half hours between the two cities.
Have you booked your cruise yet?
Does the cruise line offer transport to Venice?

On weekdays there are fast train connections (2 hr. 6 min.) departing Trieste Centrale at 8:16 AM and 9:16 AM. The next departure is at 10:58, and that one takes 3 hr. 1 min. A day trip to Venice from Trieste is going to subject you to a lot of travel time and leave not much time in Venice itself. Venice is wonderful, but just being there for a few hours in the middle of the day will make you one of the hordes of daytrippers trudging the mobbed streets along the route taken by virtually every such visitor, from the train station to the Rialto Bridge and market and onward to San Marco.

I wonder how early you'll be able to get off the ship in Trieste and what time you need to be back. Unless you have a really long port day there, I'd be very tempted to enjoy Trieste rather than spending so much time traveling back and forth to Venice. I know Trieste isn't Venice, and I love Venice--I recently spent nine days there. But a cruise stop in Trieste is just not a great opportunity to see Venice, a city that really calls for more than one full day of exploration.

I've never cruised, so I don't know how much time advantage there would be from signing up for the (no doubt expensive) cruise-ship excursion to Venice. I gather the cruise-excursion folks are able to get off the boat first, but I don't know whether that's a substantial advantage at Trieste.

Trieste is one of the ports actually used by the cruise ships which say they are starting or ending at Venice, but aren’t allowed into the lagoon. So maybe this cruise ends “at Venice” (actually Trieste) and they just want to head there to stay for a few days post-cruise.

Good point. That would be a completely different matter. In that case, I'd head to Trieste Central and take the train.

You will have to get a taxi from the cruise port to Trieste Centrale
railway station,

Or take the local bus that stops close to the cruise dock. Or if the weather is nice, walk.

If you want to go to Venice, don't go to Mestre. Take the train all the way to Venezia Santa Lucia train station, which is in Venice proper - from there you can walk everywhere or take vaporetto boats (like water buses on the canals).

You could also take a bus but again not to Mestre - just take it to Piazzale Roma in Venice, which is fairly close to the train station as well, on Venice proper. In Trieste, the bus and train stations are adjacent.

If the scheduling is better, you could take a bus one way and a train the other, though personally, I would take the train both ways if I had a choice. The exit from the Santa Lucia train station in Venice out to the Grand Canal is quite dramatic and a memorable introduction to Venice.