Is the quote from Act 1, Scene 3 "time and the hour runs through the roughest day" a metaphor or a simile?
I do not think that this is either a metaphor or a simile. The words "time," "the hour," and "the roughest day," are not being used to refer to anything else. Both similes and metaphors use a noun to refer to some other noun. We say that someone "ran like the wind" to show how fast he ran or Romeo says "Juliet is the sun" to show how radiant and beautiful she seems to him. This is not happening in this quote. Instead, I would say that this could be an example of personification if we say that "time and the hour" are given the ability that living things have to run.
If you have to choose between metaphor and simile, though, this is clearly a metaphor. That is because similes always have the words "like" or "as" in them and this quote does not.