To echo DiverScience, the most common allergy to cats is to the dander. Therefore, an antibacterial spray won't do anything about this problem, but repeated vacuuming will, particularly on carpeted surfaces. If the cat's spending time on a surface which can safely be bleached, that will also help a lot, as will washing the bed linens frequently. If it's feasible, you could also exclude the cat from one particular room in the apartment--most helpfully, your bedroom--and breathing in there will get much easier as time goes on and you continue to clean regularly. This is also one of the few cases where an air purifier can actually help, as most of them are relatively better at trapping things like dust and hair than they are things like smoke.
As a tangent, based on an earlier comment, I strongly urge people to use many fewer antibacterial sprays and soaps than are currently used. The overuse of antibacterials is one of the main causes of the spread of antibacterial resistance, as all of those antibiotics end up in sub-lethal doses in the water supply, providing many new chances for bacteria and fungi to evolve resistance to them. This is true of antibacterial soaps, not just antibiotics prescribed by your doctor. Further, the vast majority of bacteria are not harmful to you, but wiping out all of the benign ones can allow harmful ones to proliferate freely, as their competition is removed. Soap that doesn't have a specific antibiotic in it (most often these days, antibacterial soap will be carrying triclosan) will still kill the vast majority of bacteria on your hands, merely by the detergents rupturing the bacterial cells if you wash for at least 30 seconds.