The new Light Blues leader has been hugely influential for Ally McCoist's side this season, leading by example as the Gers start out on the road to the SPL.
Weir said: "Lee's a winner, there's no doubt about that, and at a club like Rangers that is such an important thing to have.
"When I arrived at Rangers Walter Smith said that to be a Rangers player you had to have a hunger and desire to be successful as well as a good player.
"Ally will say the same thing, you just have to look at what he has achieved. Lee is definitely in that mould and he will be immensely proud to be captain, just as I was.
"His family and friends are all Rangers fans and he has grown up supporting the club.
"So I'm sure he will enjoy it but he will take the role very seriously. There will be ups and downs and you can't help but take your work home with you sometimes when you are in that position.
"It can take over your life having that responsibility and being a figurehead but you have to enjoy it and realise how lucky you are.
"To be the captain of Rangers is a great honour and it's something you can tell the grandkids about."
During his trophy-laden five years at Ibrox, Weir cemented his place in the club's history books as he captained the Gers to a string of domestic successes, defying those who questioned Walter Smith's decision to sign the former Scotland international.
McCulloch has proven to be another shrewd Smith acquisition after making the switch from Wigan and Weir has revealed he almost became a middle man in £2.25million deal.
He said: "I joined the club prior to Lee but I knew him from Scotland before that. He would call me during the day and at night asking me to find out what was going on and trying to make it happen. He was desperate to play for Rangers."