Assuming you share the same biological parents and neither of your parents has cystic fibrosis, there is a 25% chance that you have CF, a 50% chance that you are a carrier, and a 25% chance you are neither. If you are certain you don't have CF there is some additional prob work you can do, but it's really not worth it. Basically, it comes down to you are more likely to be a carrier than to not with the additional probability work.
As was posted above, you might be able to get the blood work done through your brother's clinic, but you don't need to do it through the clinic. If you know your brother's mutations, you can get a blood test that includes them. If you don't know them, then an expanded test like Ambry would be appropriate. An internist should be able to order the exam and based on my experience with PPOs in Texas, it is covered with no difficulties under insurance.