The young Gers will host National League side Wrexham at Ibrox in the last eight of the competition, and Mayo believes the Colts have already made a statement by reaching this stage.
Speaking to RangersTV ahead of the game, he said: “It is an exciting fixture for us, I think we have done really well to come this far.
“There’s a good atmosphere we trained this morning and it was nice and sharp. Everyone is looking forward to it and there is a lot of anticipation, you can feel it ahead of a big game.
“I think given previous years we have been knocked out very early and we have obviously been more successful this year.
“I think when you look back at the games this season, we have stepped up in individual moments where we haven’t done before.
“There has probably been more individual responsibility on the pitch and in some games we have been finding that way to get the result even when the performance might not be exactly what we want it to look like.”
Mayo has recently moved up to the first-team changing room and is enjoying the experience so far.
He added:
“I am enjoying it is an experience as well and one for me to learn from and getting closer to working in the environment round there and preparing myself for that kind of environment in the future.
“I think as a player you want to put yourself up against the best and training here with the first team you really are training with some of the best players.
“When you look at the quality and experience there it can only make you better as a player going up against some of these players.”
The 19-year-old is expecting a tough match from Wrexham on Saturday and hopes a good crowd will be at Ibrox.
He continued: “It will be great, hopefully there will be a bit of noise, the support we had at the Bratislava game I thought was incredible.
“We do really appreciate the fans there and it does give you the extra lift on the pitch.
“They might look at us and think we are younger boys and we are there for the taking but that is part of the challenge and that is what we are looking forward to.
The match will kick-off at 3pm, and admission to the game is £10 for adults and £5 for concessions. Tickets can be purchased in person at The Rangers Ticket Centre.