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.