As I understand it, Chi development is not the key intent of Push Hands. That is not to say that it won't happen, but if chi is your goal, you might be better served practising forms for three hours a day rather than push hands, as you can concentrate more on breathing and relaxation without having to also sense your partner's movement. One of the beautiful things about Taijiquan is that it is both a martial art AND a moving form of Qigong (Chi Kung).
As for time, that really is up to the natural conditions of Chi in your body. If you've never done anything, it will take a long time. If you're in good physical condition, but new to Taiji, it will take less time. If you've been practising Taiji (properly) for ten years, you probably have all the chi you're going to need. Unless you want to learn Iron Shirt technique, concentrating on Chi development is more work than necessary, since the proper practise of the art serves the natural internal circulation very well.