[DB]

Dang, I’m in my 40’s and I don’t know what it means...

[onyx]

Straight shot, no ice.

[Lancey Howard]

Okay, before I Google it I am going to put forward my guess: I believe it means, “without ice”.
Okay, back in a sec...

[Lancey Howard]

Here is the definition of "neat", per Webster's New World College Dictionary:

"neat": unmixed with anything; undiluted; straight: said esp. of liquor drunk without a mixer or chaser

Hmm..
I think most people for the past 30 years or so (I'm in my 50s) have mostly used the word "straight" to describe a drink with nothing added. I'm guessing "neat" was the common way to say "straight" back in the 40s and 50s - - you know, during Frank Davis's heyday.

[afraidfortherepublic]

Dang, I’m in my 40’s and I don’t know what it means...

Heck, I'm not going to say how old I was before I knew what the word "neat" meant in this context, but it was just in the last couple of years. I don't know anyone who uses it.