Hacker Alex Hope said he uncovered Mr Abbott's details from his Qantas boarding pass in just 45 minutes.
He then spent months attempting to contact Mr Abbott to alert him of the security breach.
Qantas said it had now updated its cyber security protocols.
Mr Abbott posted an image of a boarding pass for his flight from Sydney to Tokyo on 21 March on his Instagram account, thanking the crew.
Mr Hope said he received a message from a friend daring him to hack the former prime minister as they had recently been discussing the dangers of posting your boarding pass online.
The hacker explained in a blog post published on Wednesday that he was able to find Mr Abbott's information because his booking reference was printed on the boarding pass.
He was then able to log in to Mr Abbott's booking and search through HTML code to find his passport number and phone number. The code also included conversations with Qantas staff about Mr Abbott.
"I had Tony Abbott's passport number, phone number and weird Qantas messages about him. I was the only one who knew I had these," Mr Hope said in a blog post.
"Anyone who saw that Instagram post could also have them. I felt like I had to like, tell someone about this. Someone with like, responsibilities. Someone with an email signature."