The Spurs manager was Brannagh's guest at a Halloween showing of his production of King Lear at London's Wyndham Theatre, which is being directed by the filmmaker and thespian, who also plays the title character.
“Sir Kenneth is a big Spurs fan and was kind enough to invite my wife and I to the theatre the other night," Postecoglou said.
“We don’t get too many nights away from the kids and adult time so we enjoyed it.
“I did have to swot up on King Lear. It wasn’t part of the curriculum at Prahran High where I grew up! I had to do some research.
“You always learn from life and the appreciation of seeing people excel at what they do is inspiring. You walk out of there thinking, 'I don’t know how they do it'.
“He was good after it. I caught up with him briefly, he was very kind with his time.
“He was all over how it’s going and gave me some little bits of advice, which I’ll ignore like he’ll ignore my advice on acting, mate!
“It was good. It was much appreciated."
At the end of the play, Lear dies from grief after learning of the death of his daughter, and Postecoglou added: "There was a fair bit going on in that play, mate! I definitely want a better ending that Sir Kenneth had, for sure!"
Meanwhile, Postecoglou, whose side face Chelsea on Monday night, says he is conscious that his own example is the most important factor in changing the culture at Spurs.
Postecoglou joined the club in the summer with a remit to not only improve results and performances but perform a full cultural reset, and he has said he is creating "a new environment".
Asked how, the Australian said: “It’s a process. A lot of it is just behaviours and particularly my behaviour. When I’m the new person in the building, people are looking at me a lot.
“So what I do, what I say, how I behave is a fairly good indicator of the environment I want. I’ve said before if I walk past someone and don’t say hello, people notice. If I’m not here first thing in the morning, people notice.
“If I’m not really passionate about what I do, people will notice.
“If I don’t come in here every day feeling pretty blessed that I’m doing something I love, people will notice.
“That’s your starting point. I think most people will get a fairly good indicator of what i want by just seeing me.
“Then you try to get like-minded people around you, the staff report in, whether that’s new staff or existing staff, and hopefully those dominoes fall fairly quickly.
“Then people have a choice of whether they want to be a part of that or not.
“It is a process but I always feel it begins and ends with me. Regardless of what kind of environment I want to set, if I’m not showing or believing in those behaviours, if it’s just something I want to do artificially, if I want to be strict around time and how we talk and make rules, then I don’t think people are going to respond if they don’t see me actually doing it.”