This sentence is phrased awkwardly. I thought Russians used the 12-hour clock?
No, the standard clock is 24 hr. That said, in everyday life people would still say "час" or "час дня" (without any numerals!) when referring to 13:00, unless some potential ambiguity is involved.
So this sentence would usually sound as "Сейчас двенадцать, а не час".