Does cockup means making a mess of a job? Is it rude to use because of the word cock? Kindly cite me some examples to help me understand it.
It's not rude in BrE- it's slang, but not the same as cock alone.
It would definitely raise eyebrows, as said above, in the U.S. We seem quite afraid of this word. It feels vaguely pornographic. While the f-word is still considered very vulgar by many and impolite by just about all, it's ubiquitous. You'd cause less of a ruckus saying "f'ed up" than you would "cocked up" here.
I'm not a teacher, but I write for a living. Please don't ask me about 2nd conditionals, but I'm a safe bet for what reads well in (American) English.