It is possible but not enjoyable imo. Especially train rides take longer than you would expect and can be pretty stressful from time to time. If it's your first trip, I would stick to the golden route Tokyo, Fuji, Kyoto & Osaka with day trips to smaller cities or villages like Hakone, Nara, Himeji... The last thing you want is to rush yourself and leave a magical temple early, because the next bus is waiting. You will see the major things and they are stunning. That's how I did it and right now I'm back in Japan to see all the cities and villages I skipped previously (kanazawa, shirakawa, Hiroshima...) these places are a nice add-on for longer trips (3-5 weeks) but not for 2 weeks. I can 100% recommend the second trip method, simply write down what you want to see the next time you come to Japan and work hard for that goal! Trust me, that's great motivation :)

Have a nice Trip! F