Yes, you can easily get to Southampton by booking a coach transfer. There are a few companies that operate such services from different locations in the UK. It is worth checking the cruise line’s website to see which coach company the cruise line is cooperating with.
Cruise Connect for P&O, Cunard, Princess brands
Eaveasway Travel for RCI, Celebrity, Fred Olsen, P&O, and Princess Cruises.
National Express is also offering coach services to Southampton departing from London Victoria Coach Station. Please note that sometimes cruise lines offer free coach travel as part of their promotion (please check at the time of booking if this is being offered and available). You will still need to contact the coach company that the cruise line partnered with to book your seat before the departure.