Nice work on this!
Are my descriptions over the top? Not enough?
I really liked the descriptiveness of the setting. The opening lines did good to hook me in, and made it easy to envision the unpleasantness of the situation.
By subtly mentioning Five Guys specifically early on, it also helps add to the setting by giving people a rougher timeline of when this is taking place. There's still plenty of mystery about what happened and how much time has passed, but now I know it's based around the more modern era.
You have the character reflect back on what things used to be, like their favorite library, which helps us learn a little more about them. Now we know: they have an appreciation for reading. You also hint at them being a front-line soldier, and a kid, in a (good) casual way.
As Shadowulf said, I also like the tense atmosphere. Descriptive-heavy stories that rely more on the actions of the character and what they see, rather than leaning too much on talking, makes for a more "ambient" kind of experience. It fits the setting well, and I found it to be really enjoyable as an introduction to your story.
Is my writing too cliche?
The main cliche that distracted me was the "This can’t possibly get any worse", only to be followed by an expected “You had to ask if things could get any worse?”
I'm guilty of catching myself using it too, and usually try to drop it, or attempt to put something clever to fill the same void. It was entertaining the first dozen times, but you see it used so often these days. I usually don't recommend people use it unless it's being used as part of a joke, where its usage is being poked fun at.
The rest of it seemed fine. Reminded me a bit of Fallout, which isn't a bad thing. Post-apocalyptic zombie stuff have been pretty popular these days, but I didn't find anything boring about your story. It's more about how you approach the setting as a writer, than about the setting itself, so it didn't feel too cliche with how you approached it.
Are there any noticeable grammar mistakes?
I'm pretty lax on grammar. General rule for me is: as long as the word flow is pretty clean, and things like there/their/they're are used correctly, I'm usually not distracted enough to notice any really minor errors. Speaking from experience: if you spend too long obsessing over too many tiny things, then suddenly you're not spending time writing (or reading) anymore.
Overall it reads good, but this spot would benefit from a comma to be a little easier to digest:
"I tried to look inside but it was now too dark to see anything more than the velvet drapes and the shadows of rows upon rows of bookshelves that I remembered walking up and down trying to pick a single book out of all the choices."
Any suggestions for improving my writing style?
No real suggestions so far. I'd have to see more of the story before I might have better feedback. I like the descriptiveness of your writing, as it keeps me interested without feeling too boring or over-the-top.