What is the best way to get to the cruise port from Heathrow? Also, hotel recommendations?
Other than a private car service the best way will be National Express Bus from Heathrow to Southampton Coach Port then a taxi (or even walk to several of the terminals)- https://www.nationalexpress.com/en
Services leave from Terminal 4, Terminal 5, and the Central Bus Station (linked by underground passages to Terminals 2 and 3) broadly every hour and take around 2 hours.
You need to know which terminal at Southampton as there are 5 cruise terminals- 3 in a line in the docks closest to the City Centre (the higher numbered docks) and two others at different locations in the further out docks (the lower numbered docks)