Definitely go for CAT. Don't worry about your age, in the long run, benefits of doing MBA will outweigh your age.
Don't go for PM online bootcamps from anywhere. In my experience, it is not worth it. Companies don't looks for certifications, instead they looks for real world experience in handling products.
Plus having an MBA degree will open up lot of avenues for you like getting into sales, marketing, management consulting, equity research, financial analysts and obviously product management. Based on your interest, you can choose any career path.
And I hope you have thought through the real reason for you for getting into PM roles. Hopefully it is not just the salary packages and the fancy titles.
Source: MBA+ 7 years of PM