The best way to get an asthma diagnosis is to take a video of your cat into the vet. Otherwise, you're depending on the cat having an asthma attack while at the vet's office, and that can take several visits.
It is important that you go to the vet for a diagnosis and treatment. According to Richard Goldstein, DVM, associate professor of small animal medicine at Cornell University’s College of Veterinary Medicine,
veterinary counsel should be sought at the earliest stage. “If the
initial signs of dyspnea [breathing difficulty] are ignored,” he points out, “the condition
can rapidly progress to the more severe stage. Then the cat may die
unless emergency treatment is immediately obtained.”
There are two types of treatment that I'm aware of, your vet may choose one or both depending on the severity of asthma. Mild cases may just require the use of an inhaler (with a bronchodiliator) for every attack. More severe cases will use a combination of regular inhaler use and systematic steroids (pills).
The goal with the medications is to reduce the swelling in the airways to allow the cat to breathe easier. Inhalers (if tolerated) are generally preferred over pills as steroids can cause a number of side effects in cats when used long term.
When administering an inhaler to a cat, you'll be given a chamber with a mask. The mask fits over the cat's nose and mouth, and the inhaler is seated on the opposite end of the chamber. There's a little flap on the chamber so you can tell that the cat is breathing, and you'll count the number of breaths to ensure that your cat has breathed enough of the drug (either your vet or chamber manufacturer will give you instructions on the number of breaths).
There are two main types of masks that I've seen used with cats: a small children's mask and a special veterinary mask (Aerokat). For our cat, the veterinary mask provided a much better fit (so we were sure he was breathing the medication), but you have to order it online. Children's masks/chambers are available by prescription at any pharmacy.
I've seen folks take a couple of weeks to get their cat used to the inhaler, but my cats are used to me poking them, trimming their nails, etc, and don't protest too much, so we just plunked it on him. If you have a cat who does not tolerate being restrained or messed with, then you will want to take a few weeks to gradually accustom him to it (with lots of treats and praise).