Brookhaven is on the North/South line, and the Airport Station is South at the end of that line. So you typically just need to get on the Southbound train and ride it all the way to the Airport. Most times it's directly there, but if they're doing maintenance or at some odd hours they might have people exit at the Lindbergh Station a few stops south, in which case you would just exit the train and wait for the next Southbound one - no biggie.
MARTA - Metropolitan Atlanta Rapid Transit Authority is MARTA's web site with all info/hours/routes and if you don't find exactly what you need there, they have a customer service number you can call, too.
I'm originally from Charleston, WV by the way - enjoy your trip.
