Bear with me as a ramble for a bit - in addition it's about 10 years since I've been there, so I'm sure that lots has changed and I'd never remember the names of hotels/bars I went to.
Staying in the country - I visited Oshima, it's one of the Islands off the Tokyo coast. I'm not necessarily recommending that you visit there. I enjoyed my trip there, but I really don't think that you need to get out to the country to see another side of life. And I would be a fan of getting out and about when I travel. It was our local izakaya or even some random izakaya* when traveling around learnt about Japan
I know it's well known, etc... and not small, but
Kyoto really is beautiful and captured my imagination. I've promised myself that I'll go back there. It is a big city, but with so many nooks and crannies that you can get lost in. You can head out to Nara which was only about an hour on the train. Yes it's touristy, but it really is special. There's so much to see and do. You could spend a month in Kyoto and only touch on it. I stayed in a small Ryokan with the tatami mat floors and the owner was very friendly. I would recommend trying to find somewhere small and personal versus the large hotels of Tokyo (I've stayed in both, and I think you will get more of the local experience in the smaller place) Bear in mind in many of these smaller places the owners may not speak any English - I had enough Japanese to get by at that stage, but I'm sure you'd find a way with a guide book to point at!
My favourite city there was Hiroshima, it's modern, beautifully rebuilt after the bomb, interesting nightlife, etc... But that's a bit far to be heading to if you're only there for 4 days.
Yes Fuji San is impressive. I never climbed it, many did, but I'm not sure how feasible that is in January. I understand what you're saying about getting out of the city - but really I'm not so sure in this case. It's notoriously shy also, you can stay near by and it be hidden with cloud for the whole time.
is a day trip from Tokyo but you could stay there over night. There's a bunch of temples, etc... there's a park with an impressive waterfall (likely to be frozen when you're there) Look up, sometime in Jan there was an ice festival if I remember correctly (that may have just been that year). I made it to Nikko in all 4 seasons and always enjoyed it. Yes it touristy (but don't forget most of the tourists are Japanese, besides you will be a tourist also!).
You'll be tired. If I was there for 4 days - I'd probably spend spend my arrival/departure day in Tokyo and the rest in Kyoto. Failing that I'd spend a the time in Tokyo and pop up to Nikko for the day (maybe stay over night at a local onsen* / hot spring).
If you do decide to travel down to Koyto it might be worth getting a Rail pass - you need to buy them outside the country, ie here before you leave. Check the prices and compare to local prices.
If you're out of Tokyo consider a trip to one of the
onsens. Many of these are out doors. If memory serves me correctly, there are some up near Nikko. But make sure you read up on the correct etiquette first.
Avoid roppongi unless you feel the need to meet other foreigners. I'm sure that http://www.paddyfoleystokyo.com/ (Paddy Foleys) is still good for a pint, but for 4 days go find the small Izakaya's and select food from the
plastic models outside.
*Izakaya - Japanese bar
*Ryokan - Local Inn. Note these can be very expensive, but we stayed in a B&B/hostel type that was very reasonable.
*Onsen - hot springs, you sit in these and soak in them. It's very important to read the basics first. They are generally unisex - but not always
Generally you wear nothing in there, you're given a small (very small) towel to help protect any modesty. Make sure to follow protocol on washing yourself first.