Arteta, who played under Moyes at Everton, won the FA Cup and Community Shield within months of taking the reins at the Emirates Stadium but presided over an underwhelming eighth-placed finish last season.
They were rock-bottom of the Premier League after three games this term before an upturn in fortunes that has put them two points behind West Ham in fourth ahead of their showdown in north London on Wednesday evening.
Arteta is back under the microscope following his decision to strip Pierre-Emerick Aubameyang of the Arsenal captaincy but while Moyes would not be drawn on the news, the Scot is incredulous at some of the barracking his former midfielder has had to contend with.
Moyes said: “It was incredible Mikel won a couple of trophies. I thought he started his career really well with an FA Cup and Charity Shield, not a bad start to your managerial career.
“He’s having to readjust, change things around, get it the way he wants it and he’s in that process. He has a good team at the moment, they’ve bought some good young players.
“I didn’t quite agree with a lot of the criticism that was about, about Arsenal.”
Moyes, who will celebrate his 600th Premier League game as a manager on Wednesday, has ushered West Ham into the Champions League spots after 16 matches, proving a sixth-placed finish in the 2020-21 campaign was no fluke.
While Moyes is wary of the teams below him, he is adamant his focus is on catching the runaway top three of Manchester City, Liverpool and Chelsea, who sit third, eight points clear of the Hammers.
Moyes said: “We’ve been in a strong position, we’ll try to stay there as long as we can.
“We’re hoping to try to get to third if we can, we’re not looking behind, we’re trying to see if we can catch the teams above us, but obviously we’re well aware of the strengths of the teams below us.
“We’re in a great position. We’re going into this game ahead of Arsenal, we finished last season above Arsenal which for us was a big achievement and we want to try to do that again.”
However, Moyes, who hopes left-back Aaron Cresswell might be in contention again following a lower back injury, knows his West Ham side have a huge bullseye on their chests for the remainder of the season.
But a bullish Moyes, whose side drew a blank in a goalless draw at Burnley on Sunday following wins over Liverpool and Chelsea in recent weeks, added: “I’m saying ‘bring it on’.
“We’ll stand there, stick our chest out and we’ll take it on and see how we get on. We’ve no reason to shy away from any of the other teams, we feel as if we can play against them.”