We know an expat that's currently living in Tokyo and this is her suggestions on what to bring/buy from your home country as things are very expensive there, especially if you live in the foreigner district. Local will be cheaper.
Nappys/diapers
Wet wipes (the ones here are much thinner and not as wet as the ones you get in the west)
Medicine (ie cough medicine, flu medicine, paracetamol (for adults and children) aspirine etc)
Baby Food, here it is difficult to get oatmeal so if the child eats it then i would suggest to try and bring some.
Baby Snacks (biscuits etc). Here you can get but everything is very much based around rice
Deoderant
Shampoo and conditioner (no problem to get here, but its very difficult to see which is which)
Shower gel/bath gel (same as above but as yet i cant see to distinguish which is bath and which is shower)
Feminine Hygene products
Toothpaste/Toothbrushes
Cleaning products for the house, dishwasher detergent, Laundry detergent, if you use whitener/stain remover for the clothes definately bring some as they dont seem to have it here (well i havent found it yet). Washing machines dont have temperature setting so it is difficult to remove stains without scrubbing before putting in the washing machine.
Tea (very expensive)
Coffee (coffee is very expensive here so i would recommend to definately put a stock in the container)
Bed Linen/pillow cases (the beds here in Japan may differ in size to back home so best to pack some spare sheets and covers)
Breakfast cereal, except for in National Azabu choice is very limited and very expensive
Pasta
If you have special baby bottles, dummys and equipment for the baby best to try and bring a stock also as here they dont have western brands.
You can get of course everything here but as mentioned it is very expensive and also when you first arrive you dont want to have to be worrying about these things. Once you get more settled you can then spend the time to look at what is what and where to buy.