Learn Russian: В некоторых городах метро работает всю ночь.

Некоторый is a general word for "some" or "certain", used when talking about an indefinite part of a larger group (some cities, but not all). Кое-какой implies "a few specific ones that I know about but won't name right now". If you said В кое-каких городах, it would sound like you are being intentionally secretive about a few specific cities you have in mind.

Why is "всю ночь" in the accusative case here?

In Russian, the accusative case without a preposition is used to express duration, answering the question "for how long?". Because the metro is running continuously for the duration of "all night", the nominative вся ночь changes to the accusative всю ночь.

Doesn't "работает" normally mean "works"? Can it be used for public transport like the metro?

Yes! While работает means "works" for people with jobs, it is also the standard verb for machines, systems, and establishments operating or functioning. You use it to say a shop is open, a phone is functioning, or, as in this sentence, a transit system is running.

Why does "городах" end in "-ах"?

It is in the prepositional plural case because of the preposition в indicating location ("in some cities"). While the plural nominative of город is irregular (города), its prepositional plural takes the completely standard masculine ending -ах.