Make Japanese Rice: Wash Japanese rice until water runs clear. Transfer to a rice cooker with the water and let cook. Once cooked, fluff rice with a fork while drizzling, vinegar, sugar, and water over the rice. Cover and let cool. Set aside.
Trim ends off of the cucumber and carrot. Using a vegetable peeler, cut long, thin strips of carrot and cucumber. Set aside.
Place nori, shiny side down, on a bamboo sushi mat. With damp hands, press the cooled rice on the nori, leaving an inch on one end. Line cucumber slivers on top of rice and then top with a crab stick, running across. Using the bamboo mat, roll the nori into a tight roll. Repeat with the other sheets, using the carrot slivers with the sausage. Slice rolls into rounds using a clean, wet knife for each cut, and serve with mayo, soy sauce, calamansi, and wasabi.