The next day, with her landslide victory still to be confirmed, she landed at an empty desk, and headed out with council chief executive Neil Taylor for a walk around the city. "It took four hours," she says.
As it happened, there was no game of tennis between the two mayors on the Sunday, and there hasn't been since.
No draughts, no chess either, confirms Mr Yule, who is, however, seeking another term in his district.
They possibly could have done with a bit of competition of the sporting kind, because, believe it or not there hasn't been a lot of competition on the political scene.
The two quickly set up a Mayoral Forum where they and other Hawke's Bay mayors could meet regularly to discuss common issues, and they decided from the outset to not criticise each other or their cities "on the front page of the paper".
"It's been easy," says Mr Yule, who set the cat among the pigeons three years ago when he decided to push hard for a merger of councils in Hawke's Bay. "I think we've agreed on most things, except amalgamation."
So, to the next item on the agenda: "I think she'll be a big loss to Napier and Hawke's Bay. She's very straight-up, very forthright, but also very caring. She's been a very good mayor."
It's one point on which Napier's voters do also seem to agree with the mayor from the other side of the rivers, for she's been re-elected three times with record-breaking majorities.
She doesn't, "personally," see it as a popularity contest, but as support for the way the council has come together after its former troubled times, and has reached the stage where her subjects pay among the lowest rates in the country.
The council trumpets the city's minimal debt and recognition by an independent report as the most viable and economically sustainable council area in New Zealand.
In husband David Arnott's words she doesn't see why the ratepayers should pay for anything they can't afford, and there are perhaps three major projects which will define her era (although she would point out it's the support of the council that does that.) Already on the go when she was first elected was the annual collection of $48 from each ratepayer to provide for inevitable eventual sewerage system replacement.
A new plant's heading for commissioning, the cost all paid without the council being beholden to any bank, with no shackles of interest of the sort that has started to strangle some local bodies and their ratepayers in other parts of the country.
It was while still a councillor in the late 1990s that she first mooted the pathways along Napier's beachfront, amid cries of "it's a stupid idea, it'll never work".
The item was brought back to the table soon after she became mayor, and the Rotary Pathways Trust was formed to raise the money for what is now another Napier success story.
She made raising money for the new museum and art gallery in Napier a personal endeavour, again without leaving a cost on the ratepayer.
There were other things some said would never work, among them appointment of another woman, Kathie Furlong, as deputy mayor, from the outset.
Soon after being elected, she and other new mayors went to "mayor school," and she recalls: "We were told quite categorically that if you are female you don't have a female for deputy.
"But I, and the council had already agreed," she says.
"I think it's got to be the right person for the job, whether they're male or female."
They'd been together on the Napier Boys' High School board, which Mrs Arnott chaired, but it was coincidence that they both ended up on the city council.
Mrs Arnott says she had no plan - "I think the voters have the plans."
But she did want to stop the "negativity" she says was present, even on the council, and she wanted to "increase pride" in the city.
At the end of the first three years, the voters had spoken, louder than any opposition and challenges she'd endured, and four councillors were voted out at the 2004 election.
The future of Marineland was an ever-present debate which hotted-up in her second term, but she says some people had lost sight of the fact that the Department of Conservation and Government was rejecting the idea of having performing animals in captivity.
"I think 98 per cent of the people had put it to bed," she says, supported perhaps by the 2007 election, in which she posted a record majority in a Napier mayoral race, against a sole opponent campaigning on the Marineland issue.
Living in Napier since 1980, when she and her husband brought their two children home to New Zealand from England, she has also been aware throughout of the issues with trucks on Marine Parade, despite living a seeming world away in a Hospital Hill dead-end street where neighbours over the years have included MP Russell Fairbrother, former deputy Mayor and now MP Anne Tolley, and councillors Tony Brownlie, for a short-time Mark Herbert, and current councillor and new mayoral hopeful Bill Dalton.
Mrs Furlong is one who notes the unity of council that came in the Arnott era. "While her visible legacy will be the coastal pathways and the rebuild of the museum, of equal significance is the cohesive team she built and the sound financial position her council leaves the city in."
Cr Tony Jeffery says she's an example of "the very informed, effective and totally committed civic leader," the work often seen in the results rather than the effort.
The display of China's terracotta warriors collection at the new museum next year is one example, stemming from her efforts on a delegation tour of China, and determination that "it can be done."
Rob Lutter, one of four councillors in the election to replace her, says he admires Mrs Arnott's "ability to relate to so many," and her "passion, skills, dedication, humility and long hours she has given to Napier".
Another Napier mayoral candidate, Roy Sye, said she had put her "heart and soul" into being mayor of Napier.
"I think she has done a wonderful job and now she is passing the baton onto a new mayor and a new team," Mr Sye said.
Cr Faye White says: "Her dedication and passion for the city is second to none. Barbara is like marshmallow when it comes to family, but when something affects Napier people's lives now or in the future, Barbara is steely and uncompromising."
To get away from it all, she found sailing, and has crewed with Napier Sailing Club members Bob Witham and Gilbert Duncan for many years.
"I don't think she had done it before," says Mr Witham, recalling the day he invited her to have a go in the boat.
"But I think that for her it was a very good way of putting politics and everything else aside for four hours on a Saturday afternoon."
But it was more than that. They've competed in each summer for the last eight years in the national Noelex 25s championship, and won in rather painful circumstances at Nelson.
At the start of the last race, unaware they'd already sealed the series, their boat, Elderberry Wine, brushed up against another craft, with the Mayor's leg trapped between them.
She left Nelson for home the next day on crutches and with leg in plaster.
But it didn't stop her sailing, or hobbling into the office the next morning and carrying on with the job of Mayor of Napier.
