Great name. I have dealt with mint rust twice, both times from unknowingly bringing in an infected plant from the nursery. The first time I tried trimming it out, bagging and binning all the debris, and a bunch of fungicides that my research told me would help. Some of it worked to an extent, but it would always come back later in the season, so it became another thing that had to be watched for and dealt with on otherwise maintenance-free plants.

I never tried baking soda. I'd reckon that it would probably slow it down to an extent, but it's not going to kill the fungus.

I eventually followed one of the methods used in agriculture. Cut the plant down to the ground, and bury it at least 5 centimetres deep. Bag and bin all the debris and off-cuts, or bury them just as deeply. The mint comes back really beautifully and it's completely cured because the fungus is only able to move short distances inside the plant and it cannot reproduce underground.

It's pretty harsh, but it kills off that population permanently, so you never have to worry about it coming back unless it's reintroduced from somewhere else.

The only requirement is that the plant must have developed rhizomes. If yours is a single-stem bush that doesn't have any rhizomes then it won't work with that. The rhizome is needed to survive the burial.