You could try for the Hakone Loop, all or part.

http://www.japan-guide.com/e/e5209.html

Hakone Onsen, can't say which may have an unimpeded view of Mt Fuji. But from listing this one does, Hakone Green Plaza Hotel, I haven't stayed here but we stayed at the old Fujiya.

http://www.japan-guide.com/e/e5209.html

One option that you might consider in case of time or weather constraints is the Ooedo Monogatari Onsen park in Odaiba
http://www.ooedoonsen.jp/higaeri/english/index.html

My wife and I stopped here after touring the Science Museum and it was a lot of fun and relaxing. Close to transport so if the weather makes a Fuji view impossible you can use that travel time for another activity.

I am curious about why you would spend 3 nights in a Narita Airport hotel. Just fyi, I think there is an onsen in Narita town.

You are asking about a high-speed train. Do you specifically want to ride the shinkansen (bullet train) as part of the trip or do you just want a quick way to get there?

When is the trip? Hakone can get busy on the weekend. I took the bus from Moto-Hakone back to Yumoto and there was a long line of cars coming into the park as it was the beginning of the weekend. But you can, and probably would, avoid the road. Still, if it is a holiday weekend I am supposing that the onsen could be very busy.

I am using www.hyperdia.com to get the travel info below.

Referring to the map in the japan-guide link that kalihiwai2 gave:

There are two ways to get there: Odawara in the east and Gotemba in the west.

The obvious way to get to the area would be by JR train to Odawara. From Narita Airport JR station take the Narita Express to Shinagawa, change to a Kodama shinkansen to Odawara. That would be under two hours. From Odawara you can take the railway (orange on the map) to Gora then the cablecar and ropeway to Odaka and walk to the Hotel Green mentioned above. I don't know the travel time for that part but am guessing it is within the 2-hour ballpark, probably closer to 1 hour.

To Gotemba, the map shows JR and highway bus access. The JR options provided by hyperdia from NRT go via Odawara. The green arrow north from Gotemba on the map is JR pointing towards Shinjuku, I think. That route can be made to work but would be more complicated and would involve the busiest train station in the world (Shinjuku) and would still require the bus. An option is the Narita Express to Shinjuku and then the bus, from the bus station across from that busiest railway station. It might even stop at the Green hotel but worst case is you ride it to Togendai and then the ropeway. Using that bus might make the trip cheaper but, also, because of traffic just that much longer in time.

All that is to say that the best route is via Odawara. The Narita Expres to Kodama shinkansen would cost 6000 JPY each way. There are other ways to get there that would be less expensive and maybe not much more time. But consider 2 hours to be the minimum time to Odawara and 6000 yen the maximum cost.

That travel time from Odawara to Green hotel is a WAG, a guess. Forgot to mention that Mt Fuji (Fuji-san), not shown on the map, is to the west of Lake Ashi (Ashi-ko). And, for all I know, maybe there is a tourist bus to Hakone from your hotel or one nearby or the airport.

For your other day trip, into Tokyo, it would be in the area of 90 minutes + 1400 to 60 minutes + 3000 yen, depending on how and where you go. Those are rail options. Another way would be a limo bus from the airport to a hotel in the area you want to visit: 120 minutes and 3000 yen each way to Ginza, for example.