Both could be correct. "At the end of the day" means you wouldn't be finished until the end of the day. "By the end of the day" just gives a deadline, so you could finish earlier.

Let me give this a try: (1) I will finish my work BY the end of the day. --BY here means not later than or before a specified time. --therefore, the sentence would mean that you will finish work anytime, BEFORE the end of the day. (2) I will finish my work AT the end of the day. --AT here means the exact time. --therefore, the sentence would mean that you will finish work AT the end of the day, no more, no less.

"I am going to finish my work by the end of the day" means that you will finish your work before the end of the day. "I am going to finish my work at the end of the day" means that you will not finish it before the end of the day. Both are correct depending on what you're trying to get across. (I will going is not correct grammar.) Yes, you can substitute any of those words for the word "day".