What should I call a person who backs me up when I am away from the office?should it call: my cover or my back up or my subsitute or my delegate?
I would most likely call this person my substitute, although some of the others would be okay to use. I would certainly not use the word "delegate."
We tend to use the word "substitute" to denote someone who comes in to fill in for a person. You could use the phrase "back up" for this as well, though substitute is more formal.
In places I have worked, the term "cover" is used to indicate someone from the same workplace who comes over to take your job while you are off at lunch or some other very short-term thing (less than a whole day). You might say that you are going to "cover" for someone while they go on lunch.
Overall, though, I'd think that "substitute" would be the best choice here.