Except for those "service apartments" that cost almost as much as a hotel, I've never heard of furnished apartments (though I admit I've never looked for anything in or near Roppongi myself ).

I've also heard that it costs more to ship furniture overseas than to buy it here so unless you have some pieces you don't want to part with, it's probably not economical to ship stuff over. You could consider leasing if you're only here for a while. Both times I was shipped over to Japan I dealt with a company called Tokyo Lease (a Google search will yield quite a bit of information on the company). If you're here for a year or less, leasing should be less expensive than either buying or shipping. And they take care of both delivery and pick-up after you're gone.

If you're planning to be here longer term, they also sell used furniture, which ends up even cheaper. In my case, after a couple years I was officially transferred to the local office and lost all housing benefits -- at which time I was able to make a sweet deal with Tokyo Lease to buy the few pieces of theirs that I really liked for pennies on the dollar (like a couple of full-size bookcases for $10 each that are still in use today). If you're here for the long haul, I'd also suggest buying only the bare minimum you need to survive and then watching the free classifieds for used stuff being sold by ex-pats leaving the country. You can pick up some good deals that way and a lot of the stuff has been used only a short time (of course, that depends on what you can negotiate with whomover is picking up the tab for the relocation, I suppose).

And, unless your stay is really short, I'd stop looking in Roppongi. For whatever you can afford there, you can double it by accepting as little as a 1/2 hour commute. Unless, of course, you can't get enough of expensive concrete high-rises packed in like sardines. Or if you have an income to rival that of Dave Spector and can afford to live in a penthouse !