Learn Hardcore Russian: Через минуту я отвечу в чате.

With через meaning in / after (a duration of time), Russian uses the accusative to express how much time will pass before something happens.
So: через + Accusative → через минуту, через час, через две минуты, через неделю.

Does через минуту mean “in a minute” or “after a minute”?

In this kind of sentence it’s best understood as “in a minute” (i.e., after about one minute has passed). In English, “in a minute” can also be vague (“soon”), but Russian через минуту is usually more literal: one minute from now.

Can I also say через одну минуту?

Yes. через минуту is the most natural/compact. через одну минуту is more explicit and can sound slightly more emphatic (e.g., correcting someone: “No, in exactly one minute”).

Why is it я отвечу and not я отвечаю?