The things I would be concerned about are feline asthma, lung infection (like pneumonia), and possibly lung worms.
Cats with asthma tend to be wheezy sounding--similar to a person with asthma. They will open-mouth breath to help bring in more air to their lungs and may have labored breathing or have a lot of abdominal effort when they inhale. Sometimes they cough. Inhaled things like pollen, smoke, etc can bring on an asthma attack. These cats generally need steroids (which can be given as an injection) to calm down the airways and help them feel better. A chest x-ray will usually give the diagnosis.
Cats with a lung infection are going to feel poorly, in addition to having problems breathing. The appetite is likely to be decreased, the cat will be more lethargic and often have a fever. Again, a chest x-ray will probably be done and antibiotics dispensed.
Cats with lungworms will cough. If they have ruptured air sacs in their lungs from the coughing, then they can have problems breathing, too. They get lungworms from eating infected snails, lizards, and other similar prey. Sometimes you can be suspicious of lungworms from an x-ray or find the lungworm eggs in a stool sample. Often I will just treat a coughing cat with an appropriate deworming medication and see how the cat does.
Feline leukemia and/or feline AIDS can be a contributing factor to various problems in cats. It will lower the immune system and make the cat more susceptible to infection.
I would put my bets on asthma. Not sure what your vet charges, but I would anticipate an exam fee and a small x-ray. If asthma, then a steroid injection (or pills, if you can manage that), which would last at least 2 weeks would be given.