Preheat oven to 325°F. Grease and flour a 10-inch bundt pan.
In the bowl of a stand mixer, fitted with the paddle attachment, or with a hand mixer in a large bowl, cream the butter and cream cheese until smooth.
In a medium bowl, combine the sugar and lemon zest and mix together with your fingertips until the sugar is moistened and fragrant. Add the sugar mixture gradually to the butter and beat on medium speed until light fluffy, about 3 minutes.
Add eggs two at a time, beating well with each addition. Add the flour all at once and mix in. Add vanilla, lemon juice, and lemon extract and mix thoroughly.
Pour into prepared pan. Bake at 325°F for 1 hour and 25 minutes to 1 hour 30 minutes. Check for doneness at 1 hour. A toothpick inserted into center of cake will come out clean. Let cool in pan a few minutes before turning out onto a wire rack to cool completely.
While the cake is cooling, make the syrup. Bring sugar, water, and lemon juice to a boil in a small saucepan. Brush all over the cake while still warm.
Notes
Store the cake, covered, at room temperature for 3 days.