Fellow League One side Airdrieonians travel to Ibrox for a third-round tie on Friday and the Italian is confident the runaway leaders will have too much in hand over the division's bottom side.
Amoruso, who won the competition three times during his six-year spell at the club between 1997 and 2003, said: "The Scottish Cup gives Rangers a big chance to get back to the top level again.
"Tomorrow they will face a team who shouldn't be a problem to them but eventually after this round they will face teams that will put them under pressure.
"However, that's what they will have to face as they bid to get back to the top level again where they belong. The Scottish Cup is a good way to show everyone that things are going better and better for Rangers.
"This competition is very special for me. I won it three times and the last time was in 2003 when I scored the winning goal in what was my final game for the club. It was a very sad day for me but one that you must face in your career."
Amoruso, who cost Rangers £4m from Fiorentina and moved to Blackburn Rovers in a £1.5m deal two months later, insists he is struggling to understand the current uncertainty surrounding the club but is confident a return to the good times is on the horizon.
Speaking to Sky Sports News he added: "The situation at the moment is very confusing and I don't think people know exactly what is going on.
"But the club have been fantastic. Over the past couple of years it faced relegation to the third division which was very frustrating for the supporters and the players who had been playing Champions League football.
"It is not very nice to see all the fans celebrating the goals in the third divsion but that is the only way Ally McCoist and his team will get back to the Premier League where they belong.
"Hopefully things will get better. I don't know many teams in this world who would get so many supporters as this massive club and I hope the board will find the right solution for everybody as the I want to see everyone in the Rangers family much, much happier.
"Ally has been fantastic. He built a club last year in two weeks, or even less. Everything has been done in the right way since and hopefully this kind of improvement will happen day after day, year after year."