Flucloxacillin is a narrow-spectrum beta-lactam antibiotic and effective against specific families of bacteria.
So it is not a drug of choice for sebaceous cyst.
You may require broad spectrum antibiotic which may require a prescription from your doctor.
Placing a warm moist cloth (compress) over the area may help the cyst drain and heal.If you have a small inflamed cyst, your doctor may inject it with a steroid medicine that reduces swelling.
If you have recurrent outbreaks of cysts then a culture should be done to detect particular strain of bacteria and specific antibiotic for it.
In the meantime you should apply Neosporin ointment on it.
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Most of upper respiratiory tract infections are caused by viruses and antibiotics don't help.
Dicloxacillin can be helpful to treat a sinus infection caused by staphylococcus bacteria.
So it may provide relief if you have bacterial sinusitis.
I hope this helps.