The Arsenal boss said Friday he still felt "very connected" to the club where he spent six years, playing more than 200 games as a midfielder.
The punishment for Everton for breaking Premier League profitability and sustainability rules means they go into their match against Manchester United on Sunday level on four points with bottom club Burnley.
But the nine-time English champions are just two points from safety, thanks to recent wins and the poor form of the other teams at the bottom.
"It's a very difficult moment," said Arteta.
"Obviously it puts the club in a difficult position but they have been in many other difficult positions and they always find a way to get out.
"If there is a quality that I think describes that club, it is the courage and determination and the fight and they will fight against anything and I wish them the best."
The Spaniard, whose third-placed side travel to Brentford on Saturday, said news of the punishment for Everton last week took him by surprise.
"It's a big one in the middle of the season but I don't know the regulation or the whole story inside it so it's difficult for me to comment," he said.
Arteta was charged earlier this month by England's Football Association over an explosive VAR rant following his side's defeat 1-0 against Newcastle.
Howard Webb, chief operating officer at referees body PGMOL, has publicly explained the reason for the decision to allow Anthony Gordon's controversial winner.
The Arsenal boss welcomed the open communication.
"We want to improve the game, all of us - referees, managers, officials, owners, sporting directors," he said.
"We all want a better game and to have a better game we have to have freedom of speech, respectfully, in a constructive way but we have to promote that. If not, you don't get there, that's for sure."
Saturday's game against Brentford will be Arteta's 200th in charge of Arsenal and the Spaniard, who also played for the Gunners, said he was "very proud" to reach the landmark.
When asked which game was his favourite, he said: "The first one because it's a special moment to have the possibility to become the Arsenal manager. For me it was a dream come true and looking back to that first game it's something really special.
"I'm very proud, I'm very grateful. Obviously so many people around that have contributed in a massive way to be here for almost four years."
KEY NAMES BACK
Forward Gabriel Jesus and midfielder Martin Odegaard have returned from the international break fit to face Brentford, Arteta confirmed.
Jesus and Odegaard both missed Arsenal's last two league games due to injury, but Jesus played in Brazil's 1-0 loss to Argentina this week after recovering from a hamstring injury.
"We have another training session today but he's been progressing well. Hopefully (Odegaard) will be available," Arteta said.
"(Jesus) is available. He played 97 minutes in a really competitive match. We have another session. They had a really long flight and I have to see the condition of every single player to make the final call on the line up."
Arteta added that he will take a late call on the fitness of defender Ben White.
With Manchester City and Liverpool facing off on Saturday, third-placed Arsenal could reclaim top spot in the Premier League standings if they beat Brentford and the league's top two sides play out a draw.
"We watch the (City v Liverpool) game for sure as we are interested in the league and we are always on top of things," Arteta said.
"Brentford are a really tough opponent and it is a difficult place to go. You look at all the clubs who go there and they make it extremely difficult. They are really effective and well coached. It will be a tough match."
Arteta confirmed that goalkeeper Aaron Ramsdale will start Saturday's game, with David Raya, on loan from Brentford, ineligible to play against his parent club.
Ramsdale had been Arsenal's first choice in goal since joining in 2021 and played all 38 Premier League games last season, but he has been benched for the north London club's last eight league games in favour of Raya.
