Just finished a cruise on Oosterdam, which overnighted in Trieste for the last night. We did a DIY trip to Venice that afternoon.
Ship docked at 1:45pm and we were off immediately and walked about 20 mins to Trieste central train station. Easy to find, but a longer walk than I expected. Taxi would have helped but the train left at 2:16pm so we didn’t want to waste any time looking for a taxi at the port.
Took a 2:16pm train to Venice St Lucia. Trains are every hour and we bought our tickets at the station. Arrived in Venice at 4:21, lost some time figuring out the water bus system and took a water bus (number 2) the “long way” down the grand canal to st marks square. It was packed.
Missed the last opening at St marks (around 4:45-5pm) but did get into Doges palace at 5:45pm (last entry 6pm, palace closes at 7pm). Got out of doges palace close to 7pm and grabbed a light meal in Venice.
Walked the streets and headed back to the train station on foot. Caught 9pm train back to trieste, arrived 11pm. Walked back to the ship (felt safe and several others out walking in Trieste) and back on board by 11:30pm.
Got up the next morning and took the HAL bus transfer direct to VCE airport ( 2 hour bus ride).
Some thoughts - Venice is a stretch for a day trip especially with such a late arrival in Trieste. You will not be able to get into more than one site due to closing times. If we had missed the 2:16 train, we would have been on the 3:16 and likely missed the doges palace last time slot, unless we hired an (expensive) water taxi to take us straight from the station to st Marks square. If you decide to disembark on that first day, I think you would still be squeezed for time - you’d have to stop by a hotel in Venice before going to the sites.
The only ticket available in walkup at doges palace was a 25 euro multi site ticket. We could only use the doges palace portion - the other included museums were closed by the time we got out of the palace.
Train was 15 euros per person each way, water bus was 7.50 per person but I’m not sure I got the correct/cheapest ticket.
Bottom line - it is doable, and worth it to at lease see Venice and walk the canals, take a gondola ride etc. it is tough to squeeze in more than one site due to afternoon arrival in Trieste and later arrival in Venice.