Mueller won the Democratic nomination in a crowded primary and went on to defeat a Republican opponent in the 2019 South Bend mayoral election.

In both his primary and general election campaign, Mueller proposed continuing what he regarded to be positive progress made under Buttigieg, whose endorsement he carried.

[2][3] Mueller also worked in the office of Senator Maria Cantwell for four years, first as a John A. Knauss Legislative Fellow, and later as a policy advisor on energy and natural resources.

[35] Mueller expressed a view as mayor-elect that, while South Bend had made tremendous progress under Buttigieg, it was not yet entirely "back" (i.e., revived), stating, “After decades of decline, we finally turned the corner, but we are not where we want to be as a community.

[35] He declared that, in order to help the entirety of the South Bend community to experience growth, the city would need to address issues in its public safety system.

[35] Mueller also expressed his desire to create a working group consisting of people that have an interest in the relationship between South Bend's community and its police force.

[35] As mayor-elect, Mueller supported Dawn Jones to remain as South Bend city clerk, the only directly elected citywide office.

[46] On February 14, 2020, Mueller appointed Maurice "Moe" Scott as directory of community initiatives, a new position that Buttigieg had created months earlier for the 2020 budget year.

[56] By the end of 2020, St. Joseph County, in which South Bend is located, had seen a 25% increase in homelessness per the Indiana Housing & Community Development Authority (IHCDA).

[58] Mueller initially did not act to dismantle the encampment, declaring in mid-May, "It's my understanding it has remained clean and sanitary, and as safe as other places that could be outside without permanent shelter.

Being pragmatic, as long as this remains safe and sanitary and healthy, and doesn't generate its own safety or health concerns, then, while not ideal, this may be the best option we have in the near term.

[65] On July 16, 2020, Mueller vetoed a resolution passed by a 5–4 vote of the Common Council three days prior urging him to take several actions on homelessness, including declaring a "state of emergency".

[57] The month the report was released, Mueller announced that he would earmark $25,000 raised from the "Economic Development Income Tax" towards funding the creation of a permanent coordinator, and would consider asking the county and even the neighboring city of Mishawaka to assist in paying.

[57] Mueller also announced that he would be budgeting $75,000 raised from the "Economic Development Income Tax" to hire a contractor, as opposed to a salaried municipal employee, in the interim to begin putting in place some of the recommendations the group had given.

[57] Under Mueller, the city partnered with a local laundromat and the St. Joseph County Health Department to provide free laundry service for families in need and the homeless.

[74] In late May 2021, the city of South Bend began to interview candidates for the new position of "facilitator of homelessness efforts", a one-year contract job that will be given a $75,000 budget.

[75] In June 2021, Mueller announced that he was having discussions about partnering with St. Joseph County in order to create a permanent way to address homelessness in the city.

It is partnering with the South Bend Heritage Foundation, Oaklawn, Beacon Health and 1st Source in competing against other communities to secure these tax credits.

Mueller has stated that, unless he sees a persuasive case that an intermodal freight hub at the airport would have strong economic benefits, he would remain tepid on the idea of contributing municipal funding to a new station there.

In response to this, Mueller declared that he would need to look at a formal request from the group before he would comment on whether he would entertain the possibility of dedicating city funds to a study of this concept.

[86] In mid-April, Mueller also announced that the city had pledged $600,000 for CDFI groups loaning capital to small businesses not covered by federal grants.

[87][88] On May 1, Mueller and St. Joseph County's deputy health officer, Dr. Mark Fox, both criticized governor Eric Holcomb's decision to move the state towards stage two in its reopening plan.

[82] During their COVID-19 response, Mueller and St. Joseph County's leadership have often voiced disagreement over policy regarding the virus with Dave Wood, the mayor of South Bend's neighboring city of Mishawaka.

[82] Mueller advised compliance with Centers for Disease Control and Prevention recommended practices such as wearing protective masks, social distancing, and limiting gathering sizes.

[89] In October 2020, with St. Joseph County seeing a spike in cases and hospitalizations, Mueller remarked, 'It's no secret that I thought we were reopening a little too fast in May, so we never got the numbers down to a safe level.

[91] In late January 2021, South Bend launched an "innovation grant" program to compensate restaurants for operational changes they have undergone during the pandemic.

Mueller voiced his belief that, in order to have a successfully managed police force, the city needed to offer its officers "competitive salaries".

[129] On March 30, 2020, Mueller, after months of avoiding taking a position, voiced his support for a referendum that, if approved, would see an increase in property taxes to fund $220 million for South Bend's schools over the next eight years.

"[114] In a late-May statement responding to the George Floyd Protests, Mueller declared, "I hear the anger and empathize with those who face systemic injustice."

Change won't happen overnight, but I'm confident we can overcome whatever challenges we face together.”[102] By the time of his mayoral run, Mueller was dating Kellye Mitros, an administrative assistant at Notre Dame Law School.