Only open during Oyster season (months with "r" in the them) and limited times even then -- but this place centrally located uptown is THE place to go for a real local oyster experience.
Shucking in New Orleans is fast and dirty - so expect some grit - it really is normal. The oysters are HUGE, sweet and fresh and only $11/dozen (up from year's past but now a cheap price).
Also try to fried seafood - this place is a good example of how New Orleans takes frying to another level. The fried clab claws, oysters and catfish were all great and the shrimp were the best. A very good gumbo once you spice it up.
The real experience you are looking for if you are an adventurous tourist.