The translation above is fine, but "...a problem with the car" would be very commonly used, at least in the US East Coast/Mid-Atlantic region. In most cases, the meaning would be understood to be "with my car," though it could easily be used in the sense of, say, "I have a problem with the car that I rented for this trip, and I'm not happy!"
If there's nothing in the Czech sentence that would make "I have some problem with the car" completely wrong, I will add it.
The Czech sentence is clearly singular. And, at least in the US, a construction like "I have some problem with my car" is an accepted use. It may help to think of it as "I have some (kind of) problem with my car." And if it makes you feel any better, "I have A problem with my car" is also accepted.