Hallgrimsson speaks for first time in Qatar
It is Hallgrimsson's first job since leaving the Iceland post and guiding his country to their first World Cup finals earlier this year, and he admitted even he was "surprised" to end up in the Gulf.
He said he had been in talks with German and Major League Soccer sides -- Stuttgart and Vancouver Whitecaps -- before being approached by Al-Arabi, but had no regrets about choosing a team from the World Cup 2022 host nation.
"I had some opportunities, some discussions with clubs but nothing that was more interesting than this one," he said.
The 51-year-old has signed a one-and-half-year contract.
Hallgrimsson added: "This is a big, big challenge, a big club, that had historical success in the past and they would like to have that past again."
Al-Arabi are currently mid-table in the 12-team Qatar Stars League at the halfway stage.
Starved of success in recent years, the last of their seven Qatari league titles was back in 1997.
Hallgrimsson said he wanted the owners to give him time to improve the side, in a league where coaches are changed regularly and ruthlessly.
"They know that this is not an instant job, this is not buying success for today, this is a long-term project to build up success for the future," he said.
The Icelander is the club's eighteenth managerial appointment since 2010.
"A lot of people have warned me that this is a hot seat so hopefully a guy from Iceland can cool it a little.
"All the things from the president, from the club, the conversations, is that they will support me and I feel that they really mean it.
"Sacking coaches two times a year doesn't get you anywhere."