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Gary managed the Championship side between 2005 and 2010, leading them to promotion from League One and to the second-tier play-off final.
Lee, 34, played under his dad for four years at Ashton Gate.
"He's seen me laughing, seen me crying and he's seen me on the phone all day dealing with agents," said Gary.
"He had a front-row seat to a manager's plight," the 60-year-old told BBC Points West. "The good times, the bad times and how you have to get through it all - and unbelievably it looks like he still wants to be a manager, even after all of that!"
Aged 31, Lee became the youngest manager in the Football League when he took over Oldham Athletic in 2013.
His only other managerial role was at fellow third-tier side Barnsley, who he left to join the Robins on Saturday.
But Gary says Lee, who was in charge of the Latics when they were considering signing convicted rapist Ched Evans, has enough experience to deliver in his new post.
"There was the Ched Evans situation at Oldham and lots of other situations since," said Gary, manager at National League leaders Cheltenham Town. "He has coped with all of that and come through it.
"He is a learner and a listener. He likes to go everywhere where he can learn something - he has been to Barcelona and to Real Madrid.
"He will do it his way and I think his way is a good way. I think it is an exciting time for Lee.
"He is his own man now and that is important. I loved my time at Bristol City. We had our success and it always holds happy memories.
"But Lee knows everything he needs to know but if he asks for advice I will give it to him."
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New head coach Lee Johnson "knows everything he needs to know" to be a success at Bristol City, according to his father Gary Johnson.