Searching on "nebulized antibiotics", I found this article, which
suggests that it was not proved to work:

http://www.aetna.com/cpb/data/CPBA0593.html

<snip>
The SinuNEB (SinusPharmacy) is a device that nebulizes antibiotics,
anti-fungals, and medications for the treatment of sinusitis.
According to the manufacturer of the SinuNEB, administration of
anti-infectives via nebulization directly to the lining of the sinuses
results in a more rapid response, greater effectiveness, reduced
re-infection, and fewer side effects than oral or intravenous
anti-infective administration. The brand name for the unique
formulations used by SinusPharmacy in their treatment of sinusitis is
AdhesENT. These special formulations supposedly can increase the
medication's adherence in the sinus cavities, and improve the
effectiveness of the aerosolized treatment by allowing delivery to the
disease site.

However, there are insufficient published clinical studies to support
these claims. SinusPharmacy (Scheinberg, et al., 2002) reported on the
results of an uncontrolled study of nebulized antibiotics 41 patients
with sinusitis, reported an “excellent” or “good” outcome in 34
patients (82%) after 3 to 6 weeks of treatment.. Vaughan & Carvahlo
(2002) reported on a retrospective chart review of patients with
chronic sinusitis who were treated with nebulized antibiotics. The
investigators reported clearing of the initial bacteria that the
nebulized antibiotic was directed against in 28 patients, clearing of
infection and a new infection with a different organism in 10
patients, clearing and re-infection with the same organism in six
patients, and persistent infection in six patients. Because these were
not prospective, randomized studies, no firm conclusions about the
effectiveness of nebulized antibiotics in sinusitis can be drawn from
these studies.

The only published randomized clinical study of nebulized antibiotics
found that nebulized antibiotics had no significant effect. A
randomized clinical study from Desrosiers, et al. (2001) involving 20
patients with chronic, refractory sinusitis found no clinically
significant difference in effectiveness between nebulized
tobramycin-saline solution and nebulized saline. These results lead
the authors to conclude that “addition of tobramycin [to saline
nebulizer] appears to be of minimal benefit.”

No published guidelines on sinusitis management from leading
professional medical organizations discuss any role for nebulized
antibiotics. Thus, aerosolized anti-infectives are considered
experimental and investigational for the treatment of sinusitis.
...