In two hours you can visit every souvenir shop inside the gate and get some lunch also. Just walking around the lake only takes 15 minutes. Add a few more minutes if you want to go more slowly, have some tea at the tea house, and look into the shrine.Next to the garden is Colonel Janes house, the real "American Samurai" who came to Japan in the early Meiji to train and teach western sciences to men in Kumamoto. He also brought the "mikan" (mandarin orange) from California to give unemployed samurai an income.
The fastest way to get there is by street car. Taxis are sometimes much slower because of traffic.