He stands at the helm of a government trying to deliver ambitious, in many cases visionary, changes to the city (the tram, the lowering of Parkes Way to bring development to the lakefront, a new public hospital, a nation-leading commitment to renewable energy, astonishing in its scale), and he does so at a time of budget nightmare.

It's not easy to think of a more gentle word for Barr's budget, forecast to plunge this financial year to a $770 million deficit. It's the worst number by a very, very long way that the territory has been forced to stare at wide-eyed, and one that greatly surpasses the deficit of any other state or territory (other than Western Australia, which is running a deficit of $1.29 billion - although on per capita basis, the ACT's deficit is worse than Western Australia's also, more than four times as bad).

The picture to be unveiled on Tuesday will look better for 2014-15 than the forecast suggests, but worse for 2015-16. Barr confirmed on Friday there would be a significant improvement this year at the expense of next year's deficit, as a result of timing being pushed back for the Fluffy buyback - with much of the Fluffy cost now in 2015-16.

Mr Fluffy is certainly by far the biggest hit to the deficit this year and next, but even before the Fluffy decision, last June the government was forecasting a 2014-15 deficit of $333 million – huge by Canberra standards.

Labor's deficits began in 2012-13, and things haven't recovered, although Barr was still talking last week about charting "a credible path to surplus".

We'll discover on Tuesday how the figures are stacking up, but while Barr has made much of the ACT's fundamentals being sound – low debt, a strong balance sheet and a triple A stable credit rating – he will surely have been giving more than a little thought to fighting his first election as chief minister next year with such ugly numbers to sell.

The numbers are far from the only challenge, perhaps far from the biggest challenge.

It is by no means an easy thing for Jeremy Hanson's Liberal team to take government in Canberra, not only because the electorate is left leaning, but also because he has to do not just a little bit better but really quite a bit better than Labor to win, given the likelihood that the Greens will keep a city seat and side with Labor. The Liberals have to beat Greens and Labor put together.

But Hanson has more than usual on his side. Labor has lost the force of personality of Jon Stanhope, and now the likeability and trustability of Katy Gallagher. It will have been 15 years in power.

Then there's the tram, the most expensive infrastructure project in Canberra's history, and surely the most controversial. The government can be in no doubt of the challenge of selling the tram to voters. Intriguingly, a number of Canberrans received a call this week from Brisbane's Reachtel Research putting questions on their voting plans. Who will you vote for and how rusted on is your vote? the poll asked. Do you support or oppose light rail? Are you more or less likely to vote for a candidate who supports light rail? Are you more or less likely to vote for light rail if it created 3500 jobs during construction?

While it's not clear yet whose poll this is - Labor and Liberal both say it's not theirs - it's a series of questions very much to the point. The government, though, will surely have to do better in justification of the tram than hanging its tram hat on construction jobs.

The tram hasn't hit the budget in a significant way to date, with spending running at about $20 million a year for the Capital Metro Agency and no reason to suspect an increase in the coming financial year. The real budget impact of the tram won't come until the end of construction in 2019 or later, when the government pays the winning consortium a likely 50 per cent of its debt, perhaps $400 million.

The government intends to pay for this by selling off a swathe of buildings, including public housing – although it must also find substantial money to build replacement public housing, no fewer than 1288 units over five years.

At the same time, the annual payments to the private operator of the tram will begin, and while the government has repeatedly refused to say how much that might be, one respected economist has suggested $80 million to $100 million a year. It's the kind of figure that will be felt.

Rates increases, too, are not an easy sell, with increase of 10 per cent year on year. With two years of that already on the resentment side of their ledgers, the patience of Canberra home owners might well be wearing thin at the likely announcement of another 10 per cent, even if it does come with reductions in stamp duty. A $1200 stamp duty saving will be welcome for house buyers, but there are perhaps seven times as many home owners than buyers, and they won't share the joy.

Barr has spent the past fortnight announcing bits and pieces of what he's calling a "suburban budget". It does have an odd resemblance to the budget of a greenkeeper or groundsman, dedicated to more mowing and maintenance of ponds and trees, bike paths and walking tracks and determined to give something to each of the town centres – a Canberra Institute of Technology campus and roadworks in Tuggeranong, and surely more to come in other suburbs. "You're going to see in this year's budget a particular focus on suburb renewal - investment in our town centres and utilising ACT government ….. institutions in order to bring life and vitality to different regions of the city," Barr said.

Look out on Tuesday, though, for the bigger picture. We can expect funding for the public housing build. The new public hospital at the University of Canberra must also be funded – although the government has refused to put a figure on the project beyond saying it will cost "well over $100 million". On Friday came funding for the City to the Lake project, albeit only $10 million, for a new lakefront park, but there is much more in the pipeline – more work on lowering Parkes Way, for which the government is seeking federal funding, reclamation of part of the lake, the development of lakefront land for sale, and a new swimming pool – a project that you might have expected to see progressed given the dramatic closures of the city pool over recent weeks to search for leaks that seem to have troubled the facility for decades. On Friday, though, Barr suggested the pool was one for his next term of government.

An announcement is also due on the new $400 million government office block to be built on London Circuit beside the ACT Assembly building. The government called expressions of interest last year and by the August close 11 firms had put their hands up for the project. They have been left hanging since, with the government yet to shortlist – or at least yet to tell us it has shortlisted. Barr says that project is not a budget build; rather it will be built by the private sector, and leased by the private sector. But given plans to have it completed in 2018, an announcement on the next step is surely imminent.

But budgets have become less than helpful on capital works spending, with the letters "NFP" making an appearance where the annual figures are supposed to go in spending tables. It stands for "not for publication", and with those dismissive letters, the government excused itself in the 2014-15 budget from disclosing the price of the new hospital, the amount to be spent on the CIT campus in Tuggeranong, the amount to be spent in the out years on the new convention centre, the cost of new court buildings, and tram funding.

These projects are excluded on the basis of commercial sensitivity, in that they involve going to the market for price and the government wants to retain the maximum competitive edge. It seems a rather arbitrary argument, given that pretty much any sizeable capital works project involves a tender.

Going into the Cabinet budget process earlier in the year, Barr signalled spending "discipline" with the aim of offsetting all new spending with savings elsewhere. But he has also insisted his budget will seek to stimulate the economy and maintain government employment and fill gaps left by the Commonwealth.

The ACT loses $130 million in GST revenue from the Commonwealth for the 2015-16 year. Barr also points to significant challenges to rein in the cost of health, with costs rising about 5 per cent a year and the Commonwealth returning to a funding model that the ACT says will strip $600 million out of its budget over the coming decade. This won't hit Tuesday's budget, when Commonwealth health funding will increase, but from 2017-18 the ACT will go backwards quite drastically, receiving less health money that year than it gets this year, the government says. A similar problem hits the education budget.

Barr has not proven an easy leader to characterise, with a persona that mixes relaxed bonhomie with a prickliness he doesn't quickly shake off. And in the same way, his budget approach is elusive. His interest in removing taxes and red tape for business, supporting innovative, youthful schemes, rejuvenating the city with new building (interests, by the way, that bring no small reminder of Kate Carnell) sit well on his shoulders. But you wouldn't pick him as the leader to be exuberantly announcing small projects for the suburbs.

Tuesday's budget might well leave Canberrans with as many questions as answers about individual projects, but it will also provide a sense of the new chief minister's economic approach - whether his inclination is to bring the numbers strongly under control, or whether he intends instead to continue defending a massive deficit on the grounds of economic stimulus, whether recent concern about the cost and availability of land in Canberra is a one that resonates strongly with him, whether he has a big project still in his pocket itching to see the light, and perhaps even a glimpse of the direction he intends to take into next year's election.