Practice three-step air punching. To do this most effectively, stand with your legs shoulder-width apart. Your fists should be at your sides, your elbows slightly bent. Imagine a person in front of you and punch him in the nose with your right fist. Then, as you draw your right fist back to your waist, with your left fist punch him in the gut. Next, with your right fist, punch him in the groin. The punches should be done in quick succession.
Now you should continue the punches by punching the imaginary person in the nose with your left fist, his gut with your right fist, his groin with your left fist. Repeat the punch sequence until you tire.
Practice five-step air punching. Five-step air punching is the same as three-step except that you don't stop in the same place. You punch the nose, the gut, the groin, the nose again, and the gut before stopping momentarily. It looks like this: Right hand punch to the nose, left hand punch to the gut, right hand punch to the groin, left hand punch to the nose, right hand to the gut---all in quick succession--and then stop for a moment before resuming. Note that you can vary this as much as you wish---in fact, after you get the three-step and five-step down, you should vary the steps.
Use a speed bag. Speed bag training will help you throw faster punches---especially in situations where your opponent is very close. The stance you must use for speed bag training is very unlike the stance in air-punching. You will need to be in a boxer's stance---your fists will be in front of your chin or nose. Punch the bag as quickly as you can. Practice as often as possible for best results.
Practice with a heavy punching bag. Throwing a faster punch won't do you much good if you hurt your hand on the first punch. When you practice fast punches with a heavy bag, you will condition your fists to cope with realistic striking surfaces. Heavy bags in addition to speed bags should certainly be part of your training equipment. If you are on a tight budget, you can improvise with a bag of child's play sand attached by sturdy rope to a tree or pole.
Do quick standard push-ups daily. Do ten at a time and do them quickly. Longer, slower sets will build muscle but will not condition the fast-twitch muscle fibers for increased speed.
Do short, quick repetitions of triceps push-ups daily. As the name indicates, triceps push-ups tone and strengthen the triceps. To do triceps push-ups, simply assume the standard push-up position, only your wrists will be facing outward and your fingers will be facing towards the fingers of the opposite hand.