Gatwick to Southampton coach: the simplest one-seat option
National Express currently operates coaches between Gatwick Airport and Southampton. Its Gatwick → Southampton route page confirms services to Southampton Coach Station, in the city centre.
For cruise passengers, the main benefit is simplicity. You avoid a railway interchange and can keep large luggage in the coach hold. National Express also publishes coach stops at Gatwick South Terminal and North Terminal, so check the exact departure point printed on your ticket.
Southampton Coach Station is central, but it is not inside the cruise port. National Express notes that taxis are available outside and also describes a frequent free bus connection toward Southampton Railway Station. If you are sailing the same day, include the final coach-station-to-cruise-terminal leg in your plan.