The biography on Wikia does a good job of explaining things:
Booker DeWitt was born on April 19, 1874. At the age of sixteen, he
was part of the 7th Cavalry Regiment of the United States Army, and
was present at the Battle of Wounded Knee. Due to his gruesome
actions at the battle, other members of his regiment gave him the
nickname "The White Injun," because of his habit of taking trophies
from his victims.
After the battle, fraught with guilt, Booker
attended a river baptism led by Preacher Witting, to be reborn a new
man and absolve his past. However, Booker rejected his baptism before
it could be completed.
By 1892, when he was eighteen, his wife had
become pregnant with their child. However, she died while giving birth
to their daughter, Anna DeWitt. Dealing with the scars of Wounded
Knee, the grief of her death, and the stress of becoming a single
father, Booker slipped into alcoholism and gambling.
Looking for work, Booker
became a Pinkerton Agent, tasked with breaking up union strike
efforts, but was discharged due to his use of extreme methods on
Following his discharge from the Pinkerton Agency, he tried to open his own
Private Detective agency, but his alcoholism and gambling addiction
remained. Eventually, Booker became indebted to individuals that, he
claimed, one would not want to owe money to.
At some point, Booker met
a mysterious man, who offered him a deal to wipe Booker's debt away in exchange for
Anna. On October 8th, 1893, Booker reluctantly handed her over, but
regretted his choice immediately.
So he had at least a year, possibly up to 18 months to get hired as a Pinkerton, get fired, and open a (presumably poor) Private Detective Agency. Not that difficult to believe, I think.
Also, to answer your original question, Booker is 19 years and 5 months old in the final scene (if the calendar on his desk is right, at least).