One of my kitties has asthma. He gets an inhaled steroid every other day, Flovent, and has a rescue inhaler for sudden coughing
attacks, albuterol. They are administered via AeroKat. Just like with humans you need to keep it under control to prevent attacks since lung damage accumulates over time, so my cat always has to have his inhaled steroid. The inhaled steroid stays more just in the lungs though so it lessens the risks of long term steroid use. Some of the steroid will get into the rest of their body though and you may see a few side effects. For my cat though, the side effects (thinning ears, slower hair growth so he can end up with thinner hair in some places) are definitely better than him not being able to breathe.
Here is the gist of the treatment: http://www.felineasthma.org/pdf/inhaled_protocol.pdf