There is an FAQ section on the Princess website about sailings from Southampton. It doesn't say anything specifically about those with pre-cruise land tours with Princess. It recommends arriving at the cruise terminal with proof of testing but says that "for fully vaccinated guests,
we will be able to provide a limited number of antigen tests that meet our requirements, at or near the terminal." A $60 pp fee will be billed to your stateroom. We're doing a DIY pre-cruise visit, but plan to do an eMed proctored test from our London hotel room before heading to Southampton. We'll order the test kits and have them in our possession prior to leaving home for our trip to the UK. I don't know if the fact that you are doing a Princess pre-cruise excursion changes any of this, but I'd come prepared with either my own test kit (for proctored testing) or another backup plan for getting a test.
https://www.princess.com/plan/cruise-with-confidence/cruise-health/frequently-asked-questions/uk-southampton-cruises/?accordion=covid-test-prior-to-boarding