A Brilliant but reckless mechanic promises his dying brother that he will save the one girl who can cure a zombie-plague before the military kill her.

You have all the correct elements. Your main character, his function, strength and flaw are all there. The goal and stakes are also clear, and your antagonists: the military, and zombies.

I think you can reword it to make it under twenty five words, and remove the dying brother part as his motivation for saving the girl is clear enough, to cure the zombie plague.

In terms of content, saving a girl from the military/zombies doesn't provide enough of a hook to get me interested. Zombie plague has been done. Maybe a more detailed description of the antagonists would help.

Yes, but what's the story about? To me, the logline as is seems like all setup. I'd like to get a sense of what the actual story is about and what hardship the mechanic must overcome to save the girl. Ok, he promises his brother, but does he save the girl by rescuing her? Helping her escape from pursuers? That kind of specific would help me envision the scope of the story.

I agree with djmc about the military killing her. Maybe re-word it so it discusses the conflict between her and the military. This is just me tweaking the logline around, and you don't have to use it. I was just giving a brief example, and it could be a bad example but we want to know why the military wants her dead.

A Brilliant but reckless mechanic promises his dying brother he will save the one girl who can cure a zombie-plague, before the military executes her for being the cause of the end of humanity.

Its a great idea...!