Yes, cats can get sinusitis. Domestic cats have two frontal and two sphenoid sinuses. The small sphenoid sinuses are named after the sphenoid bone which is located in the middle of the skull towards the front. The frontal sinuses can become infected with bacteria following an upper respiratory viral infection (URI). Sinusitis is referred to as a secondary bacterial infection.
I am not a veterinarian but the symptoms are well know to humans. They are similar for cats: a nasal discharge often from one nostril combined with sneezing and sniffing. It can cause a headache or general discomfort in the face. Headaches are hard to diagnose in cats but a cat with a headache may sit with their eyes partly closed and with their head hanging. Sometimes cats press their heads against a hard object to alleviate headaches.
Sinusitis can also cause a depressed appetite with consequential weight loss. Other causes of a cat frontal sinus infection are fungal infections (uncommonly) and an abscessed root of a top premolar tooth. Feline fungal infections (cryptococcosis) can be caused by the dust from pigeon excreta blowing into a cat. A vet will confirm sinusitis with an X-ray.
A vet will treat the condition with antibiotics which is exactly what a doctor would prescribe for human sinusitis. Sometimes surgery may be advised if the antibiotics are unsuccessful. Please talk to a vet about it. I am simply answering a question that cat owners ask of Google.
Some pages on URIs