Arsene Wenger's men ended a much talked about nine-year trophy drought by overcoming Hull City 3-2 at Wembley last month.
Prior to their cup success, Arsenal also led the Premier League for much of the season, only to fall away and ultimately finish a somewhat distant fourth behind champions Manchester City, Liverpool and Chelsea.
Adams hopes his old side can take confidence from winning the FA Cup, but feels they are unlikely to improve their league position in 2014-15.
The 47-year-old told the Daily Express: "The club came under criticism for not winning a trophy and I was one of them (who criticised the team)! It was important for them to win that (the FA Cup).
"Hopefully it will be a pivotal moment in the careers of the squad members and they can go on and win league titles.
"However, I don't think they will win the title as Chelsea and Manchester City are blowing the league apart and Manchester United will come again.
"I don't think Arsenal will finish any higher than fourth at the moment, which is a little bit sad, but they needed to win the FA Cup because they had stick from the fans."
Elaborating on why he feels Arsenal will fall short of the mark in their quest for a first Premier League crown since 2004, Adams added: "They played some remarkable football at times last season, but they do leave themselves open.
"When they come up against big teams they tend to leave themselves too open because they want to play freely and roll teams over."
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