Rinse the sushi rice under cold water until the water runs clear. Combine the rinsed rice and water in a rice cooker or pot. Cook according to the rice cooker instructions or bring to a boil, then reduce heat to low, cover, and simmer for 18-20 minutes. Remove from heat and let it sit, covered, for 10 minutes.
In a small bowl, mix rice vinegar, sugar, and salt until dissolved. Gently fold this mixture into the cooked rice. Allow the rice to cool to room temperature.
In a mixing bowl, prepare the tempura batter by combining the tempura batter mix with cold water until smooth.
Heat the vegetable oil in a deep frying pan over medium-high heat. Dip the shrimp into the tempura batter and carefully place them in the hot oil. Fry until golden brown and crispy, about 2-3 minutes. Remove and drain on paper towels.
Place a bamboo sushi mat on a flat surface. Lay one sheet of nori, shiny side down, on the mat. Wet your hands to prevent sticking, then spread about ½ cup of sushi rice evenly over the nori, leaving a 1-inch border at the top.
Arrange a few slices of avocado, cucumber, and 2-3 pieces of fried shrimp horizontally across the rice.
Using the bamboo mat, carefully roll the sushi away from you, applying gentle pressure to keep it tight. Seal the edge of the nori with a little water.
Using a sharp knife, slice the roll into 6-8 pieces. Repeat with the remaining ingredients.
Serve the rolls with soy sauce, pickled ginger, and wasabi.