Combine rice and water in a saucepan; bring to a boil. Reduce heat to low, cover, and cook until rice is tender and water is absorbed, about 20 minutes.
Combine rice vinegar, sugar, and salt in a small saucepan over low heat; stir until sugar is dissolved, 1 to 2 minutes. Pour over rice; stir until rice cools and looks dry.
Preheat oven to 400 degrees F (200 degrees C). Grease a baking sheet with cooking spray.
Arrange shrimp tempura on the baking sheet.
Bake in the preheated oven until golden and crispy, about 6 minutes per side. Cut tails off 4 of the tempura shrimp.
Cover a bamboo sushi rolling mat with plastic wrap. Lay 1 sheet of nori, shiny-side down, on the plastic wrap. Spread 1 cup rice on top using moistened fingers, leaving a 1/3-inch border. Flip nori so rice is facing the mat.
Arrange imitation crabmeat and cucumber along the bottom edge of the nori sheet. Place 2 tail-off shrimp in the center of the nori sheet. Place 2 tail-on shrimp on the end so that the tails extend over the sides of the sheet. Lift the edge of the mat and roll up nori into a tight log around filling.
Transfer sushi roll to a serving plate. Layer avocado and unagi slices on top to cover the top and sides of the roll. Slice roll into 8 pieces using a moistened knife. Repeat with remaining nori, rice, crabmeat, cucumber, avocado, and unagi to make a second roll.