Definitely check with the cruise line. If they offer air, it may be cheaper to book through them. Also check with a reliable travel agent who has access to cruise fares (ask if they have a relationship with Centrav, one such source of them). Or, if you have an American Express Platinum card, that may be another option for its International Airline Program.

For what it's worth, I am a travel agent and do have Centrav access. I am sharing this only for your information, I do not ever solicit clients from FT. Anyway, I am seeing the nonstops on United EWR-NRT and return for $5,635 as a cruise fare special, or on JFK-HND via Japan Airlines nonstop for $5,895. I checked through American Express and their fares are about $200 cheaper ($5,700) than you have found.

Also, check booking a package via Expedia. Search for flight+hotel. Be sure to select business class, then search. On the search results page there is an option to choose different check-in/check-out dates for the hotel which allows you to choose as little as one night (sometimes you can later cancel the hotel, leaving the flight intact). I did it that way and found the same United nonstop flights for $5,512.