I am not an engineering manager but I have some thoughts to share with you about this topic when it comes to what education you would need.
I would find a school with a strong business school and a strong engineering school. Major in your engineering branch of choice (looks like you are interested in Mechanical Engineering). I think you should then consider a minor in an area of business (for example: I have a BS in Industrial Engineering and a Minor in Supply Chain). This would give you a step up on other mechanical engineers applying to the same position as you. In addition, I would also consider getting an MBA as this would set you apart from other candidates even more!
As far as what your day to day will look like - that will depend on the company you work for and the specific role you are given. Work on defining what being an engineering manager means to you. If you know what you are wanting from a position it will help you narrow down your search and also help you in your interviews. Best of luck!