Learn English – Does “within an hour” mean before, after, or both

Ungodly is being used in the sense of outrageous; shocking; dreadful; insufferable, rather than invoking any of its religious meanings. So "an ungodly hour" just means "an outrageous time of day".

But how did the phrase come to have a connotation of earliness?

When people complain about an "ungodly hour", it is usually because they are forced to be awake when they would prefer to be asleep.

It's not unusual to be compelled to get up earlier than you wish. Some people must get up early every weekday just to get to work on time. And on occasion they might have to get up very early - to catch an early flight, for example.

It is far less common to be required to stay up late.

So, in most people's experience, when they find themselves talking about an "ungodly hour", it is far more likely to be an early one than a late one.

Think of five minutes referring to the time of now till now plus five minutes. You will be there in that time period: this would be the literal meaning.

So what it literally means is that you will be there before five minutes (or eight days) has expired. However, idiomatically it means I will be there in approximately five minutes or approximately eight days. Note that for longer periods (like eight days) approximately does not mean ten days. It means now plus eight days, give or take a half a day.

If you use within -- which is not the most common way to say this -- you are indicating the strict meaning of "before five minutes has expired I will be there."