час means "one o'clock" when you are telling time.
You see, in Russian you cannot shorten "один час" to "один" when you say time by the clock. We use «час» for that. For example, if you want to say "See you at 1 o'clock", you say «Увидимся в час» whereas "Увидимся в один" is unnatural and sounds quite confusing for a native speaker.
To specify whether it is a.m. or p.m., use Genitive «дня» ("in the afternoon") or «ночи» ("at night"):
- Я проснулся в час дня. = I woke up at 1 p.m. = I woke up at one in the afternoon.
- Я проснулся в час ночи. = I woke up at 1 a.m. = I woke up at one in the morning.
Most commonly, ночь refers to time from midnight to about 4 a.m. Personally, I already call 4 a.m. "morning" in Russian. It is different from English where morning is formally defined as time right after midnight. In Russian «утро» requires that there actually be a possibility of sunrise (at least in the summer) and doing morning activities.
"In an hour" = через час. For "within an hour" you might say в течение часа, but it sounds kind of weird if you know when the metro is closing specifically. In other cases, "within" in a time context might be "до". For instance, "within the week" could be translated as "до конца недели".