Barton was signed by Rangers from Burnley this summer, and he is the highest profile player to move to Scotland during the latest transfer window.
The 33-year-old is expected to have a vital role to play at Rangers, and Adam thinks that the former Newcastle United man will impress.
“He should cruise it,” Adam said, when asked whether Barton would be a hit at Rangers.
“He’s a good quality player and he’ll do well there. There’s a lot of expectation for them, because they have spent a bit of money.”
Barton made his Rangers debut earlier this week, after coming on as a substitute against Annan Athletic.
The midfielder played his part, as Mark Warburton’s side picked up a victory in the Scottish League Cup.
Barton is still searching for full fitness, as he has missed much of pre-season, but he should be back in peak condition by the time Rangers’ league season gets underway.
The Glasgow side will make their return to Scotland’s top flight with a match against Hamilton at Ibrox.