I've managed a lot of cruise turnaround calls, and rail has unfortunately never been a significant part of the plan, although we always provided free shuttles to the nearest station.
The biggest disadvantages are the need to get a bus to the station, then a train, compared to a coach which can take passengers directly from the baggage hall to their home town.
Providing a station at the cruise terminal would solve this issue, but in Southampton there are 5 cruise terminals, quite well spaced around the docks, so a large number of services would be needed. Cruises do fairly frequently depart from one cruise terminal and arrive from another, so it would be awkward if only certain terminals were options.
Having 500+ passengers arriving / departing simultaneously would be a major challenge to passenger flow in the terminal, but not impossible to manage.
We operated from a number of regional ports, and saw Southampton as being the home port for the whole of the South of England, including most of London and the Midlands, with passengers coming from the whole area. Other lines without regional homeports have passengers arriving from the whole country.
In the summer, all 5 cruise terminals are typically busy on Saturdays and Sundays, with a decent number of calls throughout the week, so scheduling trains every weekend would be practical. You'd ideally want a departure around 9am, and an arrival around noon - not ideal for train utilisation. I guess an 11am arrival / departure might just work, but would be awkward for the onboard team as cabins need to be cleared and cleaned, though fly cruises do similar with planes and make it work. Ideally you'd need similar timings to both London and Birmingham at least and from multiple terminals.
A big challenge would be a delay to the train and the number of passengers impacted. A delay to the ship is rare enough to not be a major concern.
Cost wise, it would need to be at least as good as a coach option, and the train would need to be at least as comfortable as a top end coach, considering the amount of baggage needed, this may not be easy for standard rolling stock.
Another opportunity would be provision of shore excursions, especially to London, for ships using Southampton as a port of call. This is common in Warnemunde for tours to Berlin, but is pretty rare in the UK, though I believe Southeastern have occasionally done this from Dover. Again, it really needs to run from the dockside.