The correct preposition in this case is "at", as in "at the beginning of August."
We generally use "at" with specific points in time, like "at the beginning of," "at noon," or "at midnight." The phrase "in the beginning" is used more often in a broader or metaphorical sense, such as "In the beginning, humans lived in caves."
So, for talking about a specific time like the start of a particular month, "at the beginning of August" is more appropriate.