Brown took the armband when he came on during the latter stages of Celtic's 1-0 SPL defeat by Kilmarnock and led out the team for the first time before the Scottish Cup win over Dunfermline.
"The manager pulled me in before the Kilmarnock game and told me I was going to be captain," Brown told the Celtic View.
"It is a very big honour and I'm glad he asked me. I have never captained a side before so this is my first opportunity and hopefully I'll make the most of it.
"It's good to have that responsibility and it's a great privilege to be captain of Celtic.
"The younger ones look up to you and all the fans that go week in, week out recognise you as the captain of their side.
"Now I have to do the team huddle and get the lads going before the game but, to be honest, I just do the same on the park as I normally do and quite a lot of the guys are talkers on the park as well, so I think that helps also."
Meanwhile, 18-year-old Paul Slane is focused on making the Celtic first team following his switch from Motherwell.
"I've always been a Celtic supporter and this was a chance I couldn't turn down," he said.
"It's always been my dream to play for the first team and that's my target.
"I loved my time at Motherwell but this was a big chance for me and I wanted to take it.
"There are a lot of reserve-type games arranged as well, which wasn't the case at Motherwell.
"The boys were telling me that they've played against the likes of Blackburn and Chelsea recently, so I can't wait for that kind of challenge."
Slane, who scored in a 4-1 friendly win against Queen of the South on Saturday, added: "I trained with the players for the first time last week and I was a bit nervous to start with. But as soon as I started training I felt fine and really enjoyed it.
"The likes of Robbie Keane were involved but you can't really think about who you are training with. You just have to show what you can do.
"It was great training with top-class players though. Some of the stuff Robbie was doing was unbelievable. All you can do is learn from it."