On that day is correct. Some older versions of the bible use the phrase "in that day", but it is not modern usage.

As for time, "in that time" is used within, beside, or inside, while "at that time" is a certain point in time, or some other fixed point. For example, please respond to my email by next Friday. You should be able to read the book in that time. At that time we will discuss what you think of the book.

In that time they day I guess