The former Coventry, Southampton, Celtic and Middlesbrough boss was confirmed as Craig Levein's successor this week.
Strachan emerged as the early favourite when Levein was relieved of his duties in November, following a poor start to the World Cup qualifying campaign.
His appointment was welcomed by McCoist, who himself earned 61 caps for the Scots and also worked as a coach under Walter Smith with the national team.
He said: "I'm delighted. Eventually we have Gordon in charge.
"He was the man for the job for the majority of people.
"I know Gordon reasonably well and I think he will do a great job. I'm very hopeful that Gordon can get us back on track.
"I don't think he is hungry because he has been out of the game for two years - I think Gordon is generally a hungry character in terms of wanting to do well and wanting to succeed.
"That won't have changed, I wouldn't have thought.
"I know Gordon keeps up to speed with football all over the place and he's been to games during his time out of management.
"It certainly won't be a problem.
"The one thing we will get is somebody taking the national team who will be really keen and has a desire to do well for the team and the country."
McCoist believes his Rangers players will be given the same opportunity as anyone else to stake a claim for a place in Strachan's plans, regardless of the club's Irn-Bru Third Division status.
So far, Ian Black is the only Rangers player to feature in the national team since the club was demoted to the bottom tier of Scottish football in the summer.
Lee Wallace, who has six caps, and David Templeton are among those who have performed well for the Light Blues this term, and McCoist expects Strachan to be open-minded when assessing his options.
He said: "I'm certainly not going to attempt to tell him how to pick his squad or his team.
"What I would say is, I think if our players deserve to be in the international squad, I don't see any reason why Gordon wouldn't pick them. I'm sure he would."
Meanwhile, McCoist does not believe Strachan's decision to base himself in England should be a problem.
He said: "Not for me. He should be happy where he lives and he obviously wants to live in England.
"They say the world is a small place now and so is the UK.
"Gordon can travel to games, he'll have scouts going to games. We've got players down in England and players up in Scotland.
"It's not a concern to me at all."
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