In Russian, "первый час" means "between 12 and 1", implied closer to 12. This sentence is the literal Ukrainian translation. Does it have the same meaning in Ukrainian, or does it actually mean "It's one o'clock now" as the course says?
Here're an interesting link about this topic: http://yak-my-hovorymo.wikidot.com/nerozuminnya-z-chasom
Basically, it can mean both. Ukrainian-Russian dictionary state that «о другій годині дня» can mean either «во втором часу дня» or «в два часа дня» (although many prescriptive sources insist that only the second translation is correct).
So you would never say це одна година? Would that be incorrect and awkward Ukrainian?
I wouldn't use it when telling time, but it could be used in some contexts (e.g. шістдеся́т хвили́н — це одна́ годи́на '60 minutes is 1 hour').