You can try Fuji, a traditional Japanese restsurant at Tryp Corobici Hotel near Sabana Park.
It was rated by La Nacion about a year ago as one of the top 3 authentic Japanese restaurants in SJO.
I went there a couple times now and the food was up to par with any good authentic Japanese restaurants in North America .
We had sushi, sashimi, tempura, and the rest had Japanese beef curry and rice. They all tasted fine by me, sashimi was good, even the curry.Also, they have soba on the menu. Don't know if white tuna is a local catch, it was very tasty.
The place is styled like any traditional Japanese restaurants, with varnished unstained wood, with the sushi bar at the left of the restaurant .
The place was quiet enough for intimate dinner. Great service, the only thing lacking was ticas in Kimono

Anyway,I went there just for the food and personally I found the ambience a bit lacking, as in any typical Japanese restaurants.
It's located on the second floor of the hotel at one end of the corridor. It's not cheap and some amigas were thrilled to go there.
The restaurant is long and I must say I haven't spotted any tatami rooms. Then again may be they have a couple around the corner in there, my focus was not too on the place, just food and chicas. Give them a call, may be they have tatami.
El Lamero wrote:
but a traditional Japanese place where you sit on the tatami floor and the meal is cooked right at your table.
A place where you could take a chica for a little romantic atmosphere would be a nice touch as well.
Now if you mean Shabu Shabu/Sukiyaki, I don't remember seeing that on the menu.