May be a different process in Fulton County, but my experience with a city was that you went first to the local police department and got a request for information under the freedom of information act. You filled this out and took it to city hall and filed it there. The stuff you requested then came in the mail a few weeks later.

For Fulton County Magistrate Court and State Court go to www.fultonstate.org/onlinerecords.php  
For Fulton Count Superior Court records the link is down. In the future you can google fulton county superior court case search. You can also go to the Fulton County Jail inmate search it will give a little information but sounds like you need to go through the clerk of courts (Magistrate, State or Superior all separate). Looks like the jail link is down too.

Fulton county has all case records on-line now. You can pay and get the record you are searching.

lerk of the Court for Fulton County.  Go there, get in line, have fun... or try browsing through their website.   http://www.fcclk.org  

or try a paid site like Legal Dockets. I've used this site before. Not too bad to deal with.  Beats going to the courthouse.     http://legaldockets.com/georgia