This plant needs to clear several major hurdles:
Rate of Oxygen Production
Option A: Ridiculous rate of oxygen production
Oxygen is "generated" by plants taking carbon dioxide and water and releasing purified O2 gas as a byproduct of sugar production. As in Nepene's answer, the efficiency of conversion must be significantly higher than that of ordinary green matter to sustain a human being. No matter; these plants have extremely high surface area through their fractal leaves and also displace high volumes of air through micro vibrations of tympanic structures throughout their leaves that act as high efficiency fans to allow this gas exchange to occur at an accelerated rate, allowing each plant to do the work of a hundred normal plants. This gives rise to a telltale whooshing sound as air is moved around and through the plant's "green lung" that is as audible and palpable as human breathing when reduced to a comparably sized opening.
Option B: Air purification
Alternatively, the plants don't need to produce all of the oxygen necessary to sustain the carrier, but produce earthly amounts of oxygen and instead very efficiently filter out chemical, biological and nuclear contaminants from the atmosphere. (The plants as you say are carried around as air purifiers. The air contains enough oxygen to sustain life but is not breathable due to high concentrations of toxins.)
Stupendous energy intake
All plants require an energy source. The more CO2 it processes or the more matter it filters from the atmosphere, the more energy it must consume. Most terrestrial plant life uses photosynthesis. The Sun is dim in this post-apocalyptic world, so photonic energy harvests are poor. Instead, these plants leverage a strange sort of symbiosis with their toxic environment by harnessing nuclear fission of radioactive particles, or by consuming chemical toxins from their environs as reagents to an energy-producing lifecycle that allows them to undertake the demanding work of purifying air and producing plant sugars. The downside is that once they are removed from this hostile world, they will wither almost to the point of dormancy or die from the shock.
In the case of producing unearthly amounts of oxygen, The plant would also have to be growing our outputting solid or liquid byproducts at an enormous rate.
The plant (or the human) needs to be able to get rid of excess carbon. Trees are made of about 50% carbon, much of which is directly absorbed from the atmosphere during the air purification/sugar production process. A typical human being exhales over 1 kg of the stuff every day. Over the course of a week or two, the plant would double its mass if it were performing full duty producing O2 and sugars, making it unmovable within a matter of days.
If it is purifying the putrid air from its surround, it would also accumulate nontrivial chemical deposits which would need to be purged regularly.
Through a biological miracle, this particular plant regularly extrudes and sheds its carbon reserves into sugar bubbles and/or shingles of a peat- or coal-like substance that break off easily and can also be consumed as fuel directly by burning--which unfortunately consumes oxygen, so the atmospheric oxygen content is decreasing with time, requiring either greater efficiency or a greater number of plants to balance the environment.
The plant also requires hydration. Water being too heavy to carry around in addition to the plant, either the plant or its containment vessel must do very well at hanging onto water. Cacti and other desert plants are designed with minimal surface area so as to retain moisture. Unfortunately this plant makes the opposite design choice by sacrificing water retention for rapid air filtration and so loses hydration ten times as rapidly as any ordinary plant. This makes for an exceedingly delicate balance--if the plant is ever exposed to raw atmosphere it will dry out rapidly unless it is kept at close to 100% humidity. For this reason it must be kept either fully or almost completely enclosed in a bubble that traps humidity on the inside, maintaining a jungle or rainforest climate inside the jar. You could get sophisticated here and say that it harvests moisture somehow from hydrogen gas vents through a complex reverse electrolysis process, or by humans feeding the plant chunks of some kind of fat, but I think the humidity control vessel offers a simpler and more credible explanation. Processing sugar and producing oxygen also consumes water--but maybe the plant, the human or a symbiotic machine or organism is able to liberate and recycle the water from the sugars, and produce pure carbon (hence the carbon shingles).
Of all these requirements, energy consumption might be the most challenging.