Solskjaer is in his second full season with the Manchester-based club and have lost four semi-finals since he took over from Jose Mourinho.
United face a defining moment in their season this week, with the second leg of their Europa League last 16 at AC Milan on Thursday sitting at 1-1 from the first leg, and the FA Cup quarter-final against Leicester City on Sunday.
"The FA Cup has become critical to this football club this season. The League Cup and FA Cup has been the starting point for Manchester United teams over the last 15 to 20 years,” Neville said in his podcast.
"If Ole Gunnar Solskjaer is going to be successful at this club, he not only needs to finish second in the Premier League, but he has to win a trophy. That would be seen as real progress. Then you're attracting players next season to try and catch Manchester City.
"I think it's a brilliant week. Beating Manchester City the way they did, and winning on Sunday. If you'd have said to Solskjaer at the start of the week you'll beat Manchester City, beat West Ham and draw 1-1 with AC Milan, he'd have snapped your hand off. Because going into that Manchester derby, they were under real pressure.
"If they'd lost to City, there's a real pressure building on this West Ham game. I think they are now a lot more comfortable going into the final weeks. This was a big, big week.
