They survive... in a way.
Technically the marsupials living back then in the middle Eocene are all dead. Deader than a door nail, deader than a dead parrot used in a Monty Python skit.
So no they didn't survive.
They did however have descendants. Of which we do call some of them Microbiotherians. The others we call Eometatheria.
The distinction is precisely the setup you gave.
On the one hand, Microbiotherians (which currently live in South America) look almost exactly like those distant and very dead ancestral marsupials. This is because they share the same kind of lush tropical rainforest environment. The environment has maintained the same selective pressures, and having already struck on a good solution to those pressures by the middle Eocene, the descendants have stayed very similar.
On the other hand, Eometatherians where hit hard by the Azolla Event. The landscape changed and thus the selective pressures changed. Some evolved into macro fauna as large as any rhino, some developed high speed locomotion (kangaroos), some are in the process of evolving fly organs (sugar gliders), and others have converged back to forms not to dissimilar to Microbiotherians (marsupial possums), although the journey has changed them.
The slower continental drift is the only unanswered issue. The problem is that the Australidelphia are completely wiped out in Antarctica. So there is definitely an extinction zone.
The question is whether or not the Australidelphia managed to get far enough North that the Azula event was changing the environment slow enough that they could ride out the problems.
Given this map of the Eocene, and moving the continent roughly half way down toward the antarctic. The environment does indeed look grim. Most of Australia would have been within the arctic circle, or very close to it. However there is land to the north, the island of New Guinea.
As Shadowzee has pointed out in their answer, the north of Australia would not have been too far below present day Tasmania. Temperatures would theoretically be some 5-10 degrees cooler.
Also do not forget the the island of New Guinea currently has marsupials (proving that it could be traversed to) and would be roughly where Tasmania is presently under your scenario. Granted it would not be covered in a rainforest, but an aboreal forest, or temperate Forest at the height of the Azolla event would not be out of the question.
Therefor we can easily infer that New Guinea would have marsupials roughly equivalent to present day Tasmania. A macro predator the size of a small-medium dog, numerous herbivore species specialised for grass, and leaves. And also a few marsupial possums.
The northern reaches of Australia would be somewhat rougher, but would be comparable to Tasmania's present day Mountains. The forest would be thinner, and the grasslands would be seasonal tundra. Unfortunately this would probably mean that marsupial possums would not survive, not unless they evolved some form of hibernation, or squirrel like lardering.
There would be some space for an omnivoure/small predator marsupial specializing in the summer on birds, fish, amphibians burying kills for winter. If omnivorous it might survive the winter on carrion, barks, and pine needles. Presuming that Australia was far enough north, and evolution quick enough, to keep them out of the extinction zone.
Presuming Australia was completely marsupial free because of the Azolla event.
This would not have stayed the same, as Australia would still be drifting north. The Iceage that came dropped sea levels significantly, and if the environment in Northern Australia was still to inhospitable, when the environment improved (and sea levels returned) there would still be a significant and closely linked chain of islands that could facilitate Island hopping back to Australia from New Guinea when conditions improved.
The story of the Australian Mega Fauna would have been slightly delayed, and somewhat more bird orientated, but eventually the marsupials would have come back with similar basic adaptations to current day Eometatherians.
So in a sense the descendants survive, we just don't call them Microbiotherians. In another sense no the Microbiotherians are extinct within the realm of Australia.