How can I get better at cross country?

This is going to be my first year running cross country (or running distance competitively at all). I ran track this past spring, but I was a sprinter. Right now, I can run around a 28 minute 5k, but by mid-season, I want to be able to run a 22-24 minute 5k. Is there any possible way to do this? How long will it take, and what would be a good running schedule? I have mild exercise induced asthma.