Everyone is giving you some excellent advice. The JR rail pass is an excellent deal since two way fares on the bullet train (shinkansen) will cost about the same as the rail pass. For me unfortunately you can't get one and use it with a work visa. You have to have the tourist visa to buy it. One big thing though is that you must purchase a voucher before you come to Japan because according to the site I looked at before I came you can't purchase the pass inside Japan. You trade in the voucher once you arrive.
For coming through customs at the airport you will need to know the name of the hotel you will be staying at to fill in on the embarkation card or whatever they call it. I don't know how it is for Russians (I think you said you are Russian) but coming through customs is fairly easy for US/UK/Aussies/etcetera. If you don't know where you will be staying just make something up.
As far as sight seeing in Tokyo I could show you a few places you can go. Yoyogi park on Sunday afternoon is where you will find the Nihonjin (Japanese people) in costume. The towers everyone has told you about in Shinjuku is Tokyo Tocho. These are the Tokyo Metropolitan offices and admission to the viewing deck is free. This is the best place to go first because the ground level of the towers has a Tokyo/Japan tourist center. They have free maps of Tokyo and these helpful flyers that have a map for every district in Tokyo and the top attractions in the district. The maps are really good and after following some of them, the sights they suggest are not bad either. Tokyo Tower (the Eiffel of Tokyo) is best at night and so is Sensoji and Kaminari-mon in my opinion. Because one, they are all lit up and pretty at night. Two, there are a zillion people (Japanese and Gaijin "foreigners") pouring through Kaminari-mon and Senso-ji during the day. These two are right next to each other. As far as the Ueno Royal Museum, it is a real bummer you are going to miss the Salvador Dali exhibition since it closes on the 6th of Jan. But for the latest happenings in Tokyo I use
:: Metropolis Tokyo :: Japan's No. 1 English Magazine XMAS SPECIAL which is an English e-zine or something.
The rest of Japan I have not seen yet because I have only been here three months and am working most of the time. I think you said you are coming mid-Jan. and maybe I will have a weekend to help you out, but this is the peak of snowboard season so my weekends may be a bit full.
One more thing, for ease of traveling the train lines you can check out
Welcome to Jorudan Co.,Ltd. which gives you times and line transfers. The fares are not always accurate meaning sometimes it is cheaper. I highly reccomend Japan Atlas: A bilingual guide published by Kodansha International. This is about the best map I have found unless you speakuth Nihongo. Best wishes and happy holidays and as I tell my students, Ganbatte Kudasaine (try hard/good luck). Ja ne...