Aw, we got engaged on a trip to Tokyo, so it will always be a romantic place to me. What a great city, though, and tons to do. A lot of it will depend on what you and your fiance like to do, though. I’d suggest getting one of the lonely planet-type guidbooks and starting from there. A good website to check out (and for any place that you go on a trip to) is virtual tourist (virtualtourist.com, i think–maybe google it).
We organized our trip by picking one or two districts to visit per day. This worked really well, as there’s plenty to do in every major district and trying to do more than that would have been really, really exhausting. It’s a huge city and even just getting from one district to the next can take a lot of train rides and walking.
I’m not sure if they do a big honeymoon-type-suite (or if that’s what you’re looking for) but the hotel we stayed at–Hotel Sunroute Shinjuku–was beautiful, very accommodating, reasonably priced and in a great central location (about two blocks from Shinjuku Station, which you will pass through A LOT if you’re traveling around the city). I highly recommend them. Also, if you’re coming in from Narita, definitely take the "Airport Limosine" to the city-proper. It’s actually just a charter bus, but it’s pretty reasonable (about $30 each way per person) and makes stops near all the major tourist hotels. It makes it very easy to get into Tokyo when you’re jet-lagged and not super used to the train system yet.
I also want to plug Asiana Airlines if you haven’t already booked your flights. We took them and they were FABULOUS. So much better than a US airline. Seriously, the best airline I’ve ever flown. They’re a Korean airline, so you usually get a layover in Seoul, so that might not be your ideal, but we actually had a great time during our layover. It’s a very fun airport and Asiana has a relaxation lounge with beds, computers, showers, massages, etc for its passengers there (even coach!) that’s just really, really nice.
As far as sites in the city to visit, I think it will really depend on your interests. I will recommend the observation deck at the Tokyo Metropolitan Government building in Shinjuku. It’s a great view of the city and it’s free. There’s also a tourist office in the building where you can pick up these handy walking tour sheets for each district in English. I would also say that Asakusa is great, as is Ueno Park and the Meiji Shrine. Shinjuku is a total kick to visit at night, as is Shibuya (where a lot of love hotels are also located). If you’re short on time, I’d say skip the imperial palace (you can’t see anything, anyway, though the grounds and the moats are pretty) and ginza, unless you’re really into high-end shopping, though it is fun to see the crazy architecture and gawk at the insanely expensive stores. This is going to be an odd suggestion, but if it’s raining, rather than visit an indoor museum or something, I’d recommend getting a pair of the transparent umbrellas they sell everywhere (every convience store and train station will immediately start selling them for about $5 at the first sight of rain) and hitting Yoyogi Park. It’ll be a lot less crowded and it is sooo spooky and beautiful there when it’s raining.
Other than that, I guess just be prepared for everything being close to Manhatten-type-prices, though you can get less expensive food and such if you look for it, and be sure to hold hands when crossing major streets. The crowds get pretty unbelievable at certain times there and it’s easy to get separated from each other. Also, don’t worry about having to read or speak Japanese. Almost all major signs will also be in English and in the tourist areas a lot of people will speak at least a little bit of English. Just learn to say "Arigato" and to bow your head a lot. Knowing a little more is helpful, but you can pretty much get away with just that because everyone there is sooo nice to tourists and so eager to be helpful. Also, don’t fear the crazy-looking train system. It’s actually pretty easy once you ride it once or twice.
Sorry for such a long post. I just had such a great time in Tokyo and hope you do too. What a neat place to have a honeymoon! Congrats!