Learn Russian: Урок продолжается час.

The ending -ся makes the verb reflexive. Here, it changes the meaning from doing an action to an object into an action happening on its own. Instead of saying someone continues the lesson (продолжает), we use продолжается to mean the lesson continues itself or goes on.

What case is the word час in here?

It is in the Accusative case, even though it looks exactly like the Nominative (dictionary) form. In Russian, when you state how long an action lasts, you use the Accusative case for the time period. Because час is an inanimate masculine noun, its Accusative form does not change from the Nominative.

Why is there no word for "for" in the Russian sentence?

In English, we say "for an hour" to show duration. In Russian, you don't need a preposition like "for" when talking about how long an action lasts. Simply putting the time word in the Accusative case (час) is enough to express "for an hour".

Is the verb продолжается imperfective or perfective?

It is imperfective. When a sentence emphasizes the ongoing duration of an action, like going on for an entire hour, Russian naturally uses an imperfective verb.