Can you get sinusitis from a cat if you are not allergic to it?

Unlikely: Allergy causes inflammation around the opening to the sinus which may cause an infection. If you aren't allergic, that is unlikely.

Sinusitis: Sinusitis in the most simple of terms means inflammation of the sinuses. Cat dander can be very irritating and even though you are not technically allergic to the cat, if the cat goes outdoors it can pick up pollens that stick to its fur and/or it could also be the quantity of cat dander you are exposed to (sleeps with you) that can cause symptoms.

Probably not: Sinusitis almost always starts with a viral rhinitis (cold) Unless the cat was sick, and you do not have specific cat allergy, it is highly unlikely that you can develop a sinusitis from a cat.