I would check with National Express
. It's the British coach operator like Greyhound in the US. There is a direct coach service available from Heathrow Airport to Southampton.
I would also check with the cruise ship line which normally also offer convenient transfer services for cruise-only-passengers.
You may also check with Intercruises
. It's the agent of CUNARD who is arranging the transfers from/to Airport and Southampton. As PRINCESS is managing CUNARD I am quite sure that they help you as well.
I have no idea how much you would like to spend for a night:
Express by Holiday Inn and Novotel as budget hotel
Holiday Inn as mid-price hotel
Hilton and Marriott as high-price hotels
So enjoy your cruise + Happy Holidays + Happy New Year!