Why doesn't My Xbox 360 S update the clock when connected to Xbox Live?
Solution 1:
Go to setting and check that the correct timezone is set, sometimes Xbox will default it to GMT.
Restart the console to take effect.
Comment this back if that doesn't help.
Solution 2:
Perhaps you don't have Daylight Saving mode on.
Set the time and date
Press the Guide button on your controller, go to Settings, and select System Settings. Select Console Settings. Select Clock. Change the clock settings.
Note: You cannot change the clock settings while you are connected to Xbox Live.Date and Time: Set the date and time.
Time Format: Choose the 12-hour or 24-hour clock format.
Time Zone: Specify your time zone.
Daylight Saving: Choose On if you want your console clock to adjust automatically for Daylight Saving Time.
Source