Preheat oven to 350°F (175°C). Grease a loaf pan and line it with parchment paper.
In a bowl, rub lime zest into the sugar using your fingers until fragrant.
Add eggs and whisk until smooth and pale.
Whisk in oil, coconut milk, vanilla, and minced lemongrass.
In another bowl, mix flour, baking powder, baking soda, and salt.
Gently fold dry ingredients into the wet mixture. Do not overmix.
Pour batter into the prepared pan and smooth the top.
Bake for 45–55 minutes, until a skewer comes out with a few moist crumbs.
While the cake bakes, simmer all syrup ingredients gently for 10 minutes. Strain and keep warm.
Poke small holes all over the warm cake. Slowly pour syrup over the top.
Let the cake cool fully before slicing and serving.