2. The date of the Bar Mitzvah-The 13th day:
As soon as a child begins his 13th birthday, which completes 13 years of him living in the world and begins his 14th year, he is considered Bar Mitzvah. There is no need for him to wait a full day after his birthday, or until his time of birth on that day, to be considered Bar Mitzvah.
For other questions relating to dates of the birthday, such as leap years, or born Bein Hashmashos, or the 30th of Kisleiv. See here
 Admur 53:13; M”A 53:12; 235:4; Bach 53; Shach C.M. 35:1; Kaf Hachaim 53:53; 235:10; See Encyclopedia Talmudit Vol 4 Erech Bar Mitzvah; Nitei Gavriel chapters 1-3; Chapter ?? for various details relating to establishing birthdays; See Likkutei Dibburim p. 353 [Hebrew] that the Mittler Rebbe received his Aliyah on Shabbos, and not on Thursday, and that some of the Lamdanim explained that this is because the Alter Rebbe wanted to wait 24 hours from his time of birth. However, the Alter Rebbe himself later clarified that this was because the time of Mincha of Shabbos Is the most auspicious time to receive an Aliyah.
Other opinions: Some Poskim rule that he must wait until the time of day that he was born to be considered of bar mitzvah age. [Divrei Chamudos on Rosh Nida 5:5; See Encyclopedia Talmudit Vol 4 Erech Bar Mitzvah]