Correct Option (2)

To determine the day of the week for a specific date, the concept of "odd days" is utilized. The reference point is given as January 1, 2001, which was a Monday.

The calculation proceeds as follows:

Number of years: The period spans from 2001 to 2099, encompassing 99 years.
Leap years: Within this 99-year period (2001-2099), the leap years are 2004, 2008, ..., 2096. The number of leap years is $(2096 - 2004) / 4 + 1 = 23 + 1 = 24$ .
Non-leap years: The number of non-leap years is $99 - 24 = 75$ .
Odd days from years: Each leap year contributes 2 odd days, and each non-leap year contributes 1 odd day. Total odd days from these 99 years = $(24 \times 2) + (75 \times 1) = 48 + 75 = 123$ .
Odd days from months in 2099 (up to June 1):
- January (31 days): 3 odd days
- February (28 days, as 2099 is not a leap year): 0 odd days
- March (31 days): 3 odd days
- April (30 days): 2 odd days
- May (31 days): 3 odd days
- June (up to 1st day): 1 odd day
Total odd days from months = 3+0+3+2+3+1=12.
Total odd days: Sum of odd days from years and months = $123 + 12 = 135$ .
Net odd days: To find the net change in day, divide the total odd days by 7 and take the remainder. $135 \div 7 = 19$ with a remainder of $2$ . Thus, there are 2 net odd days.

Since January 1, 2001, was a Monday, June 1, 2099, will be Monday + 2 days = Wednesday.

Now, we determine which date in June 2099 is a Sunday:

June 1 = Wednesday
June 2 = Thursday
June 3 = Friday
June 4 = Saturday
June 5 = Sunday

Therefore, June 5, 2099, is a Sunday.

Incorrect Options:

Option 1 (4): If June 1, 2099, is Wednesday, then June 4, 2099, would be a Saturday.
Option 3 (6): If June 1, 2099, is Wednesday, then June 6, 2099, would be a Monday.
Option 4 (7): If June 1, 2099, is Wednesday, then June 7, 2099, would be a Tuesday.