The Toffees were a long way short of their best on Saturday as they suffered a 3-0 defeat at home to Wigan Athletic to deny them a day out at Wembley.
Neville admits it is important for David Moyes' men to put that disappointment behind them as quickly as possible and focus their full attention on the battle to secure a top-five finish in the Premier League.
He told Sky Sports News HD: "The Champions League place is in the distance, a European place, we would take that now. It's what we have been going for all season, making sure that next season there is European football being played at Everton. That's our aim and remains our aim.
"If we can finish in the top four then so be it, but at the moment our short-term aim is to win next Saturday against Man City, to recover and bounce back and show a reaction that we didn't get on Saturday against Wigan and to attack the last eight or nine games of the season.
"It's the time where we need to stand up for ourselves. We had a disappointing February, picked up our form going into March and then last week was a major setback for us. But this team, this club, has got major character and I fancy us to bounce back."
Neville concedes that he and his Everton team-mates let themselves down against Wigan.
He added: "I think it was more than disappointing, it was a disastrous weekend for us. It was a disastrous performance and there is no hiding away from it. Personally, I was embarrassed by my own performance, it wasn't good enough as a captain and all the lads feel the same.
"We can't hide away from the fact that we have to take our medicine now, take the criticism that comes our way. We really felt this was going to be our year in the FA Cup but we were beaten on the day by a team that was better than us.
"Sometimes you can lose cup ties and you can be unlucky, but there was no unlucky about it on Saturday, we were well and truly beaten and we need to put that right. We have got two months left in the season and we have to bounce back and get a reaction. That's what the manager wants and I'm confident in the group of players that I share a dressing room with that we will do that."
Speculation regarding Moyes' long-term future at Goodison Park has resurfaced on the back of a cup humbling, but Neville hopes the Scot can be persuaded to prolong an 11-year stint on Merseyside.
He said: "That's something that only the gaffer can answer. I have loved every minute of my time playing with him.
"He has brought all of the players in the squad to the club and we all want him to stay, but that's up to the manager. We aren't using that as an excuse for Saturday's performance -anyone that does, that's a bit of a cop out."