Preheat oven to 350 degrees F (175 degrees C). Line a round cake pan with parchment paper.
Beat 1 1/4 cup sugar and butter together in a bowl using an electric mixer until smooth; beat in eggs, 1 at a time, beating well before adding the next egg. Stir flour, lemon zest, and baking powder into creamed butter mixture until batter is smooth; pour into the prepared pan.
Bake in the preheated oven until a knife inserted in the center comes out clean, 45 to 55 minutes.
Combine 1/2 cup sugar and lemon juice in a saucepan over medium heat; cook and stir until sugar is dissolved, about 5 minutes.
Poke holes into the top of the warm cake and pour syrup over cake. Cool slightly before cutting.