This idiom or expression has to do with:

Having a chance or an opportunity.

A. Why are you directing such a small budget movie?
B. This is my first shot at directing any kind of movie, and I’m not gonna miss it!!

This idiom or expression means:

A boost.
A helpful deed.
Something that gives new energy to a person who is weary emotionally, mentally, physically, financially, etc.

Q. Did you get a year-end bonus?
A. Yes, and what a shot in the arm it was! I was just about to file for bankruptcy, but now I don’t have to.

Messed up.
Screwed up.

My chances of getting a job with the university are shot. I don’t think they’ll hire me!

This is about doing something with the hope that it will work. There is no way to predict whether or not it will be successful.

We’re not having any success, so let’s talk to your dad. Maybe he can help us, maybe not. Anyway, it’s a shot in the dark.