"You weren't due up here" is casual speech. It means "You weren't due to come up here". In this phrase, "due" means "scheduled".
In the sentence "You'll play havoc with the computer getting here before your time", the verb "get" is changed to have "ing" to explain a cause or reason. This is called the present participle form. We can use the present participle instead of words like "as" or "because". So, this sentence means "You will break the computer because you got here early."