Hi welcome to CC (Cruise Critics)
I live about 190 miles from Southampton and always travel down the day before. I don't want the stress of travelling down the same day in case of road closures due to accidents.
If you could post the name of the ship we can give more specific advice, about parking, as Southampton has different terminals for different cruise lines.
Check out the Ports of Call section of CC for advice about the ports you will be visiting:
https://boards.cruisecritic.com/forum/8-europe/