As the film briefly explains, the intention to treat Alzheimer's disease in the movie is through viral gene therapy. Although the movie understandably simplifies this mechanism and its characteristics, it is an approach that has a sound theoretical basis and some evidence in practice. Several clinical trials have attempted to apply gene therapy, but the results have not yet yielded the therapeutic success as seen in the movie.
Viral gene therapy could prove to be useful in diseases or syndromes that are caused by a lack of certain proteins. A virus must be engineered that exhibits the normal capacity to invade human cells, yet none of its normally destructive properties. That virus would carry a piece of suitable genetic information (DNA) and release it inside the human cells, which would then lead to those cells producing the necessary protein(s), thereby compensating for the defects caused by the disease.
Practically, there are many problems and limitations associated with gene therapy; the movie suggests that the patient's immune system starts to reject the therapy. How this occurs is not explained in detail, but one of the drawbacks encountered in reality is the fact that delivery of genetic information to the cells is not stable; the newly added DNA is foreign, and therefore not copied into new cells, and therefore rejected; repeated treatment will be necessary. However, since the DNA carrier is a virus, with each therapy, there is a chance that the body's immune system will respond and create immunity against the carrier virus; so much that the virus is, at one point, cleared from the body before having had effect. The subject is resistant to therapy then.
In real life, the predictable problems with curing Alzheimer's disease through gene therapy would most likely not be rejection of the new DNA from brain cells. Brain tissue does not multiply as its cells no longer have the capacity to replicate. As long as new DNA is correctly delivered into those cells, rejection of it during DNA copying would not be expected to happen. The real problem is to get the new DNA into these cells in a controlled way. Getting the virus to deliver the new DNA to a select type of cells and not to others, and in optimal numbers, is the real challenge.
For the sake of story, this problem seems to have been mastered in the movie. The viral therapy ALZ-112 is administered by injection, which probably depended on the carrier virus being most efficient when entered directly into the bloodstream. When Charles becomes resistant to ALZ-112, Will suggests using a more agressive virus that can overcome the host's immune system. This treatment is called ALZ-113. Perhaps the carrier virus for this therapy can only exist as airborne particles, or within an aerosol (as opposed to dissolved in a solution), necessitating inhalation instead of injection. But keep in mind that this solution is of course primarily meant as a narrative tool to allow Caesar to infect the other apes more easily.