Do you need my exact date of arrival?

No, we do not. But it would be better if you could provide as accurate as possible for your sake, because your duration of stay in India will be 30 days from the date of arrival but your visa approval letter is only valid in 33 days from the date of issue. You can arrive and exit on any date within the duration of the letter. So, if you are not quite sure about your date of arrival, just choose some day earlier than your planned arrival date.