Is it possible to learn karate or kungfu after 32 years? how long will it take to learn its basics?

Of course! It'll be a little harder for you as you will have lost some natural flexibility (unless you do a lot of exercise so as long as you start out slow you'll be okay. Join a good school/academy and train with them. Although I'm younger than you I recently started Mixed Martial Arts (MMA) from doing nothing like it ever before. Although Ive only been going for 4 months I can touch my toes, have greater muscle definition and I'm physically stronger. Martial arts is a great fitness tool as its fun, you learn a skill and in the kicks/punches and power needed to hold it and yourself in the correct position (and balance) its a great overall workout! I'd say that it depends how much you put in but a good few months should be enough for the basics.

It is possible to pick up martial arts and any age and train. That being said, it depends on what you want to gain out of the training. If it is for physical fitness or mental well being then there are many training centers that cater to that demographic. If, on the other hand, if you are planning to turn the training into a career or into a striking based combat sport then I would urge you to reconsider. To answer your second question it can take several weeks to several months to master the basics of karate. It depends on on how much effort you put into training the basics.