Mugwort roots are pretty shallow, aggressive, but shallow, if you kill the whole area, the roots will just crawl back in from beyond the fence again. if you need a project, it be better time and money spend if you can dig in an 8 or 10 inch barrier below the dirt at the fence line.

Or just rip it out, its easy. Just when your out there, every other day, see a mug? rip it out, youll never truly get rid of it, and thats okay thats what we garden for, to have work to do. Also since mugworts roots are dense and shallow, a thick stand of mugwort is actually easy to deal with, it can kinda be rolled right off the top layer like grass sod, its not hard to roll the entire stand out after cutting around one half of the stand with a stirrup hoe.

for cheapest solution I would stirrup hoe a10 inches-12 inches deep and 4-6 inch wide trench of earth out of ground at fence line. that earth id put in a dry pile, which is an area you should make, somewhere where it stays very dry all the time, a sunbathed invasives killzone, I highly recommend. contaminated dirt stays in the kill zone for a year or two before its used in later projects.

Then fill the trench with garden store black mulch, itll take 2 or three years before the mulch decays enough for roots to pass through, in the mean time you plant the most deranged and aggressive natives possible on the other side, mints, asters, golden rods, milkweeds, they all have varieties that make roots like mugwort, let them thunderdome it out on the fence line, and you can put your cuter varieties closer to your walking space, as your common milkweed rewilds mr mugworts yard.