Preheat oven to 325 degrees F. Butter and flour a 9 inch bundt pan or grease with nonstick baking spray.
With an electric mixer, beat together the butter and cream cheese until smooth. Add the sugar and lime zest and beat until light and creamy, 3-4 minutes.
Add the eggs one at a time, mixing after each. Mix in the vanilla extract.
Add the flour and salt and mix until just until combined, scraping down the bowl a few times as needed.
Pour the batter into the prepared bundt pan and bake for 60-75 minutes, until a toothpick inserted into the center comes out clean.
Meanwhile, make the syrup by heating the lime juice and sugar until sugar is completely dissolved. While cake is still quite warm, poke all over with holes and pour the syrup over the top. Let cake sit for 15 minutes, then invert onto cake stand or serving platter and let cool completely.
To make the frosting, beat together the butter, cream cheese, powdered sugar, lime zest, and lime juice with an electric mixer. Mix until smooth.
Cover bundt cake with frosting and serve.