Prepare a grill for high heat (450° to 550°; you can hold your hand 5 in. above cooking grate only 2 to 4 seconds). Heat oil in a small saucepan over medium heat. Cook garlic and ginger until fragrant, about 2 minutes. Add jam, juices, and soy sauce and cook until simmering. Measure out 1/2 cup glaze and set remaining aside.
Wrap a prosciutto strip around a shrimp and thread onto a skewer. Alternate threading shrimp and apricots, 2 pieces each per skewer.
Cook shrimp skewers, turning as needed and brushing with reserved 1/2 cup glaze, until grill marks appear and shrimp is opaque, about 8 minutes. Sprinkle with green onions and serve with extra sauce on the side.
Note: Nutritional analysis is per serving.