Yes, of course, but he's not a prince. And he's not eligible. He's engaged.
Although his mother is Princess Caroline of Monaco and Hanover, titles in Europe normally descend from the father, not the mother, unless the mother is the reigning monarch. Stefano Casiraghi, the father of the three Casiraghi children, was untitled. Therefore, his children have no titles.
Should Caroline's brother, Prince Albert, reigning Prince of Monaco, have no children, Caroline or Andrea will succeed him. At that point, Andrea will become a prince.
But as I said, he's getting married, to his very long-term girlfriend.