I have to write a 600-word textual analysis about Of Mice and Men. Here are the instructions: explain what the book is about, what the author is trying to convey, and how successful it is as a work of art, as and indicate how it connects to American culture today. I have no idea how to plan the essay out. It seems there isn't much room to explore the ideas in this book. Any suggestions on how to structure it?

The question is written in such a way that the structure is laid out for you. Your focus should be on four major areas:
1. A brief summary of the book. Just hit the main points here.
2. What the author is trying to convey—that is, what is the overarching idea of the book in terms of themes or messages? For example, you might suggest that Steinbeck is trying to convey a message loneliness, of the shadow of the American Dream, or of familial loyalty (between George and Lennie, though they are not literal family).
3. How successful is the book as a work of art? Here, you may want to go into how you view a successful work of art in general. Does it move you, is it commercially successful, is it sustaining (valuable over a long period of time)? From there, evaluate Of Mice and Men by the standards you set. Remember, it's okay to say that a book is not successful at something if that's what you think.
4. How does the book connect to culture today? An example here might be to explore what elements of George and Lennie's American Dream are present for Americans today. Have we evolved from how Curly's wife is treated, or are there still people like Curly in the world? How is bullying present in the world today as well as in the world of the novel?
With four topics to cover in 600 words, that's about 150 words per section. You're right; it's not a lot of time to explore the ideas in depth, but if you're careful with your word choices and how much you quote to support your ideas, you should be fine. Also, remember to plan a few sentences for an introduction and conclusion, if your teacher expects that.
If you're stuck on ideas for themes and connections, I've attached a great resource for you to look at to help inspire you.