No. The Williamsburg Bridge does not connect to the FDR Drive (East River Drive) on the Manhattan side.
The FDR Drive runs along the East River waterfront; the Williamsburg Bridge lands ~0.8 km (0.5 mi) inland at Delancey/Clinton. No direct ramp exists.
South side of bridge; Manhattan entrance at Delancey St & Clinton St; Brooklyn entrance at Bedford Ave & S 5th St. Open 24/7, free.
Take Houston St exit → west to Essex St → north to Delancey St → east to bridge. Or Stanton St → Essex → Delancey. Allow 5–10 min in traffic.