"you are not allowed to wander about on your own"
This is not generally true. As a tourist with a visa I certainly walked all over on my own, and took buses and trams and the metro, no worries. But it may be true in a limited sense for you, as a cruise passenger, if the condition of your admittance on land is that you don't need a regular visa but have to follow the group leader at all times. So you follow the leader and don't stray.
Besides, a tour company wouldn't take you "off the beaten path" - they're in the business of pleasing tourists and they know (or think they know) what's good for tourists. I saw no foreigners in the markets, in the suburban shops, on the neighborhood soccer field between two factories, or hanging out with locals drinking beer outside a "produkti" shop (cheap alternative to going to a bar), but I sure met "real" Russians off the beaten path. No tour group would take you there, and in only two days there are too many fascinating major targets to see, don't miss them for some uncertainty.
Do you have a functional knowledge of Russian? Can you read the cyrillic alphabet and pronounce words? And would you understand it when people talk back at you in Russian when you ask a question? When you're dealing with people who make a living from tourism, English is usually available, for better or for worse. But as soon as you get off the "beaten path" that's all you get - Russian. So on this trip you better stick with your crowd.
If you like what you see (and it's beautiful and fascinating!), once back home you can prepare for a trip that will bring you back, on your own, for at least a week - with a regular tourist visa (not trivial!) and a basic knowledge of Russian. You'll love it, so treat this one as a convenient sampler in the short time you'll have.