An alcoholic bounty hunter is hired by a business mogul to rescue his unruly daughter, who?s missing in a zombie-ravaged New York City.

Pretty clear and concise. I only have two questions:

1. Why is the alcoholic bounty hunter chosen for the job? (Surely there's a more reliable option for the business mogul to save his daughter? Even just a non-alcoholic bounty hunter?)

2. What makes this different from a zillion other films I've seen like it? (Escape from New York/LA, Monsters come to mind instantly). What's the hook that would make me want to fund this film?

The daughter is going to be in her twenties.

I want this story takes place a month into the outbreak. Large cities are being overrun by zombies, but people living in rural secluded areas can still live relatively normal lives. The people living in rural areas still have to deal with the zombies and people taking advantage of the situation both good and bad.

Richiev: That's exactly what I was thinking! I will try to raise the stakes!

That's why I asked the age and character of the daughter.
I immediately thought of a substitute father-daughter relationship between the extraction specialist and the rescued victim.
I was thinking of the extraction specialist having no or very bad relationship with his own daughter. His daughter has just given birth, but she refuses to give him access to his grand daughter. Too many broken promises. Too many birthday missed. Lonely, depressed, and potentially dying from cancer, Going into New-York is an attempt at redemption. The money promised by the mogul would pay for the education of his grand daughter.
The mogul has the same bad relationship with his daughter. He paid for things but he was never there for moral support. Especially during the slow and painful death by cancer of her mother. The rescued daughter notices the mark left by the cancer chemiotherapy on the extraction specialist. The death of her mother from the same disease acts as a point of communication between the girl and the bounty hunter.

What I like in the TV s?rie "Walking Dead" is the depiction of the relationship, hope and despair of the characters rather than the gore part. I assumed that the script was about character driven rather than action/gore driven. Otherwise why use an alcoholic as protagonist.

Agreed, "Nothing left to lose" also leaves room for character growth.