If you arrive the day before you could stay near the airport and use the Princess transfer the next day.
I recently stayed at the Holiday Inn that’s attached to the Staybridge Suites on Bath Road. (Approx 5 mins from airport) The rooms were good size, clean and the hotel has a restaurant.
I just stood in the Taxi line to get a short ride to the hotel.
The next day,I took an uber back to the airport and followed the directions Princess supplied, It was very easy and stress free.
If you want to go into London you can take The Elizabeth line into town from the airport approx $15 per person and take the Victoria Coach Transfer that Princess Offers to Southampton