I went through the pre-julian calendar for the years 60 to 46 BCE, and taking into account Cicero's letters, which are mostly dated, and Caesar's own Commentaries, which have references to seasonal events, this is what I came back with:

Caesar's Calendar Reform

In short, what Caesar did was add a few months to the year 46 BCE so that the calendar would be again in accordance with the seasons, and regulate it so that thereafter every single year would be always 365 days long, with every fourth year one day longer.
He did not change what was past, and there was no problem with his birth year. The disparity only happened for a few years, between 50 and 46 BCE. So if he was born on the 13th of Quintilis, we are quite safe to celebrate on the 13th of July. Or the 12th, if we so choose.