It would help if we know where you were stopping at, sorry not psychic just yet.
With the exception of a few ports around the world, a cruise ship is a honeypot for taxi drivers & they will normally be queuing up along the dockside before you've had you morning cuppa. As Wayne & Lionel Blair's TV programme used to say, Give Us a Clue!......Wilba.
Sure why not....if your not going on an excursion you can do whatever you want...whatever you want within the law I guess ;).
Most cruise lines offer shuttle bus services to town centres when the port is a little way from town. Generally there is a charge but equally it isn't usually a crippling amount. Other ports are literally minutes walk from shops. Some are a little further away and my advice would be to always set a fixed price before you set foot in a taxi! Personally, I usually find asking the cruise staff generally brings forth a good answer. The entertainment team are usually quite knowledgeable and also the reception staff.
Enjoy your cruise!