• Coming of age is at 17 for wizards.
• Hogwarts 6th Year students are 16 minimum and 7th Year are 17years old.

• Sirius ran away from home at about sixteen. He had have enough.
• He camped out at the Potter's place during school holidays until he was seventeen.
• Then he got his own place having inherited gold from an uncle. After that (buying a place)  he took care of himself but he was welcome at the Potter's for Sunday lunch.

It suggests that Sirius got 17 while at Hogwarts because he sounds as though he got his place right away when he needed a roof. That would mean when he was out of school.
And his being invited to the Potter's on Sundays means he was not in school when that happened. It was either summer vacation or Xmas when he got a place. But summer is much more likely because he didn't mention "for the holidays".  So his birthday could be right before he leaves school.

However, since Sirius ran away from home at 16, he would have had to run away during summer or Xmas after starting his 6th Year. Running away for a short holiday or during his schooling would be avoidance rather than running away .

No,  he must have ran away during  the Xmas holidays of his 6th Year  or the summer after his 6th Year. However, summer is more probable because  at Xmas, he could have chosen to remain at Hogwarts.  

• But, it is still a possibility that it happened at Xmas. Then, it would mean his 17th birthday  was at least after Xmas because he was still 16 when he ran away.

• And if he ran away in summer, then his birthday more likely is at the beginning of the year because he needs "time" to turn 16. He didn't say "when I  turned 16". Therefore, we have from January to May.

Lupin and Sirius are said to be the same age, so this suggests that they were not so far apart on the calendar. Lupin was born March 10th. So if Sirius is the same age, I doubt he was born on Sept 1st which would make him 6 months older and always ahead of his friends. Another proof that his birthday must be later in the school year. Especially since James seems the youngest (see calculations at Snape's birthday).

Hogwarts students who started in 1970 were born in-between Sept.2nd 1958 and Sept. 1st 1959.
Students who started in 1971 were born in-between Sept.2nd 1959 and Sept. 1st 1960.
James still being 15 near the end of his 5th Year proves he had to be born in the exact same year as Snape and Lupin or else he would have been almost 16 or would have entered Hogwarts a year later. It is a correlation: if Snape was born in 1959, so was Lupin and James. If Snape was born in 1960, so were Lupin and James.

Also, Rowling said he'd be about 22 on Nov. 1st 1981. Not nearly 23,  about 22. Or else she would have said "about 21-22" or "about 22 -23".  Just like she did when she mentioned how old Snape was. I also believe Rowling said that because she wanted to say: "Yes, that's horrible being put in prison at such a young age and the Potters being killed while so young"  I think she wanted to say 22 because it represented the age of most of her characters at the time.

From my other calculations, Snape was 22 or 21 and 9months on Nov. 1st 1981. So if Sirius' birthday was in Sept, Oct, Nov or Dec, (near the Fall of Voldemort)  he would almost be 22-21 or just had turned 22-21. But Rowling didn't mention 21. Again, it is best to narrow our scope and situate his birthday near Lupin's (January, February, March, April, May).  

Afterwards, he went back to Hogwarts for his 7th year after living with the Potters over the summer or part of it.

Therefore, if Sirius indeed follows this pattern and was indeed 22 when he was put into prison (surely a bit before his 23rd birthday), it means that Lupin has the same age and so does Severus. Only James is younger having his birthday possibly from late June and up to August 31st. It is obvious that they were all born (Snape, Sirius, Lupin and James) in 1959 in order to be 22 by Nov. 1st 1981.