McCoist's side lost a goal in the first minute of their fifth-round showdown with United at Tannadice 12 months ago.
And the Ibrox club went on to lose 3-0 to their top-flight rivals and crash out of the competition.
But Gers recruited heavily in the summer and brought in experienced players like Cammy Bell, Jon Daly, Nicky Law and Bilel Mohsni.
And their manager is optimistic they will be able to beat Dunfermline at Ibrox this evening and go on an extended run in the compeition.
McCoist said: "I'm hopeful we're better prepared than a year ago. Obviously, we've got nearly an entirely new squad with the additions we've made.
"I'm hoping that one of the important changes is that the boys have got experience and success at cup level.
"You look at Cammy, big Jon, Ian Black, Lee McCulloch, boys like that have had varying degrees of cup success.
"Last year we had Neil Alexander and McCulloch, but I didn't feel we were as strong. I'd hope we are stronger and obviously we should be stronger with the players we've signed.
"I'd be very hopeful that would give us a better opportunity to go on a cup run. But you need to take every tie on its merits.
"We'll have to earn the right to get into the quarter-finals. If we can win tonight, it obviously takes us a step closer."
Rangers are 23 points clear of nearest challengers Dunfermline in SPFL League One and are expected to sew up the title imminently.
And, on top of that, they will play in the final of the Ramsdens Cup against Raith Rovers at Easter Road in April.
Winning the Scottish Cup for the first time since 2009 would complete an unprecedented domestic Treble for the Ibrox club.
McCoist has refused to rule out that possibility - but stressed they would need their share of luck to go all the way to Celtic Park in May.
He said: "We're in the Ramsdens final in March which gives us a really good chance of winning that.
"And you'd have to say we've got a very good chance of winning the league. So the Scottish Cup is the one that isn't just as clear-cut.
"We'd need a bit of luck. Of course we would. Most teams, if not all, need to carry that bit of luck to win a cup competition and so would we.
"I'm not saying we can't win it or we won't win it. Of course I'm not. We want to win it and we'll be doing everything we can to win it."