costs of groceries depend on many things: geographical location, city/country-side, season and of course on which kind of groceries you are referring to. Expect prices to be generally higher than in the US. What was amazing me personally the first time I went to Japan, were prices for fruits, sheer astronomical due to protective economic measures!
As for the gift: get the lady something typical from your area. Japanese do like souvenirs from abroad. That way I always fared best.
Thank you for your reply to my inquire about groceries in Japan. The reason for the inquire was that we were hoping to fix an American dinner for the family that we are visiting. I was wondering if beef was out of sight. Is it super expensive to go out to dinner, eating maybe sea food. We are already are most excited.