Learn Hardcore Russian: Я жду такси уже час. - I have been waiting for the taxi for an hour.

Yes. The verb ждать takes a direct object in the accusative. Такси is an indeclinable (borrowed) noun that ends in -и but does not change form for any case. So although it functions as the accusative object, its form stays такси in every case.

What case is час in? Shouldn’t durations use the genitive (часа/часов)?

Here час stands for “one hour” and is effectively in the accusative (since you’re measuring how long you wait). For inanimate masculine nouns, nominative and accusative singular are identical, so it looks like nominative час. Also, after the implied numeral “one,” Russian uses the singular nominative form. If you had “two hours,” you’d say два часа, and “five hours” would be пять часов.

What is the role of уже in this sentence?

Уже means “already” and emphasizes that the waiting has continued up to now. It conveys impatience or surprise that it’s been an hour without the taxi arriving.

Why is ждать in the imperfective aspect rather than a perfective form?

The imperfective ждать is used for ongoing, uncompleted actions. You are still in the process of waiting. If you wanted to say you waited (and then stopped), you’d use the perfective подождать, e.g. Я подождал такси час (I waited for the taxi for an hour).

Can I move уже час before the verb, as in Я уже час жду такси?

Yes. Russian word order is flexible. Я уже час жду такси or Уже час я жду такси are both acceptable; they just shift the emphasis slightly. The core meaning remains “I’ve already been waiting an hour for a taxi.”

Is it possible to drop такси and say simply Я жду уже час?

Context matters. If it’s clear what you’re waiting for, you can omit такси: Я жду уже час (“I’ve already been waiting an hour”). But if you need to specify, keep такси.

How would I express “I’ve been waiting for a taxi for more than an hour” or “for the whole hour”?

Use modifiers with the time word:

“more than an hour” → Я жду такси уже больше часа.
“the whole hour” → Я жду такси уже целый час.

How do I talk about different durations, like half an hour or two hours?

Use the appropriate numerical forms with genitive or accusative:

half an hour → половину часа (Я жду такси половину часа.)
two hours → два часа (Я жду такси два часа.)
five hours → пять часов (Я жду такси пять часов.)