How old is Booker in the final scene, after the credits?

Solution 1:

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 the job.

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).