Bring 1 ? cups water to a boil over high heat. Add rice and 1 teaspoon salt. Stir and cover. Reduce to low and steam until rice is very tender, about 20 minutes. Let sit an additional 5 minutes. Uncover and fluff with a fork.
Cut twelve 5-inch squares of parchment. Center one square over the cup of a 12-cup muffin tin. When rice is cool to the touch but still warm, spoon 2 rounded tablespoons into paper. Using wet fingers, press rice on the bottom, against, and up the sides of the muffin tin to form a rice cup. Repeat with remaining rice and additional pieces of parchment. You should use entire portion of steamed rice.
Combine ahi tuna, cucumber, avocado, scallions, and rice wine vinegar in a large bowl. Gently fold to combine. Divide mixture among rice cups. Top with sesame seeds and toasted nori, and serve immediately, with soy sauce, wasabi, and ginger.