It's the first ever Batman story, credited to Bob Kane. (Finger's contribution wasn't acknowledged during his lifetime.)
The artwork... well, I won't call it rubbish. It's another era, with its own standards and ideas of what was normal... but it's not good, and there were good artists in 1939. Look at Alex Raymond's work on Buck Rogers and Flash Gordon, for instance. Let's call this artwork "crude". Interestingly, though, Batman himself is already spot on. He's immediately recognisable and already hanging out with Commissioner Gordon.
He drives an ordinary car, though, not a Batmobile.
The story begins like a murder mystery, but then gets violent and 1930s-adventure-serial once Bruce is Batman. The villain captures him and sets up a deathtrap, but then leaves the room so that Batman can escape. Why, of course. SPOILER: Batman lives. The baddie's an elderly, fat businessman who'd have lost a fist fight with Lois Lane and was barely a threat for Batman even when pointing a gun at him... so our hero kills him by punching him into a tank of acid. "A fitting ending for his kind," is Batman's verdict. In that panel's background is a little death splosh.
This is funny and makes the story far more entertaining.
I read this in the DC Comics Graphic Novel Collection of Batman Hush part 1, which had it as a bonus at the back. I love that. Unsurprisingly, the story's often been retold. It got a 30th anniversary retelling in 1969, a Golden Age one in 1986 and two conflicting post-Crisis versions in 1991, published in the same issue of Detective Comics. There's also a New 52 reboot from 2014. I bet none of those had the comedy ending, though.