You're almost there. This is on Windows 7, Python 2.6.5 32bit, and GDAL 1.9.0:

 >>> from osgeo import ogr
 >>> driver = ogr.GetDriverByName("FileGDB")
 >>> ds = driver.Open(r"C:\temp\buildings.gdb", 0)
 >>> ds
 <osgeo.ogr.DataSource; proxy of <Swig Object of type 'OGRDataSourceShadow *' at 0x02BB7038> >
 >>> ds.GetLayer("buildings")
 <osgeo.ogr.Layer; proxy of <Swig Object of type 'OGRLayerShadow *' at 0x02BB7050> >
 >>> b = ds.GetLayer("buildings")
 >>> sr = b.GetSpatialRef()
 >>> sr
 <osgeo.osr.SpatialReference; proxy of <Swig Object of type 'OSRSpatialReferenceShadow *' at 0x02BB7080> >
 >>> sr.ExportToProj4()
 '+proj=utm +zone=15 +ellps=GRS80 +towgs84=0,0,0,0,0,0,0 +units=m +no_defs '
 >>>

Once you open the FGDB, then use `GetLayer` to get at your featureclass.