Hotel prices in Southampton are very variable through the summer weekend rates can be quite high.
Especially boat show week.
There are some adiquate more basic central options, good for a single night, that can have good rates.
Ibis, Premier inn, Travelodge(we use this when cheap as near pubs and restaurants)
Princess mainly use 3 terminals at Southampton and occasionally they have to use one of the other two.
A couple are walkable from some hotel options, taxi/Uber from most hotels <£10.
To check which terminal your cruise will use the following, sometimes they change so check closer to travel date.
(Our Regal soon is out from QE2 back to Ocean)
https://www.southamptonvts.co.uk/Live_Information/Shipping_Movements_and_Cruise_Ship_Schedule/Cruise_Ship_Schedule/
When using LHR we like the National Express bus as our budget option.