Preheat the oven to 350°F (175°C). Grease an 8 x 4-inch loaf pan with butter and dust lightly with flour, tapping out any excess.
In a medium bowl sift together 1 1/2 cups all-purpose flour, 1/2 tsp baking soda, and 1/2 tsp salt; set aside.
In a large bowl, beat 1/2 cup room-temperature unsalted butter with 1 cup of the sugar until light and fluffy. Add the eggs one at a time, beating well after each addition.
Alternate adding the flour mixture and 1 cup sour cream to the butter mixture, beginning and ending with the flour, until just combined. Stir in 2 tsp lime zest and 2 tbsp lime juice until incorporated.
Gently fold the blueberries into the batter, taking care not to overmix.
Pour the batter into the prepared loaf pan and smooth the top. Bake for 60–65 minutes, or until a toothpick inserted in the center comes out clean. If the top browns too quickly, tent with foil halfway through baking.
Remove the loaf from the oven and let it rest in the pan for 10 minutes, then transfer to a wire rack to cool slightly.
Meanwhile, make the lime syrup: combine 1/3 cup lime juice and 1/3 cup sugar in a small saucepan and heat until the sugar dissolves; remove from heat.
While the loaf is still warm, prick the top and sides several times with a toothpick and brush the warm syrup all over the cake. Repeat once so the cake absorbs the syrup. Let the cake cool completely on the rack.
Make the glaze by whisking 1 cup powdered sugar with 2 tbsp lime juice until smooth and pourable; adjust thickness with more powdered sugar or lime juice if needed. Drizzle and brush the glaze over the cooled loaf.
Slice and serve.