McGeady returns for Wednesday night's Champions League Group G decider.
McGeady, 26, came through the ranks at the Glasgow club before leaving for Russia in 2010.
Hoops skipper Brown, who in a bizarre incident during the pre-match press conference at Celtic Park was presented with a copy of Fyodor Dostoyevsky's 'Crime and Punishment' by a visiting journalist, expects the home fans will remember the good service the Republic of Ireland international gave to the club.
The Scottish champions will progress to the knockout stages if they get a better result against Spartak, who are consigned to the bottom of the table, than Benfica get against Barcelona at the Nou Camp.
"I think Aiden will get a good reception," said Brown. "He did really well when he was here.
"He was a joy to play alongside and he was a fans' favourite as well so I am sure they will give him that (good reception) at the start.
"But once the game starts, we are there to win it and so is he. It will be heated, fast and an exciting game to watch.
"I think we all know Aiden is a huge danger. He is a very talented boy, he is going to be a threat.
"He is enjoyable to watch and creates all sorts of problems for the defence and midfield so he is going to be one to watch."
Brown revealed he had cancelled an operation on the on-going hip problem which has dogged his season and claims it will be "no problem" to last 90 minutes against Spartak.
The Scotland midfielder said: "I am feeling good. It seems to be improving.
"I have trained five or six days and there is no pain. I got 65 minutes at the weekend (against Arbroath) so that has definitely helped.
"I am going to continue until I can't train and can't play any more.
"I have spoken to the consultant and he is happy with that so we will take it day by day, week by week. I had the operation booked it in and cancelled it because I was feeling better.
"The longer I feel fit and can play 70, 80, 90 minutes the better it is for myself and if I don't need to go under the knife I don't need to go under it."
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