Nagano

Any advise, how should I travel to Nagano upon arrival at Narita Airport?

It is a little dependent on your arrival time at Narita but I would suggest that the train is the best option. Use
http://www.hyperdia.com/cgi-english/hyperWeb.cgi
just enter Narita Aiport and Nagano in the boxes.Then date and time.
You will be most likely given two options. One will be Keisei Rwy to Nippori and a choice of two parallel lines from Nippori to Ueno and then the Nagano shinkansen. Possibly a bit daunting for a first timer and especially with luggage.It's actually quite easy if you are a confident traveller.
The other will be NEX to Tokyo and direct shinkansen from there. The only catch with that one is the NEX arrives a few levels below the main station and about as far from the shinkansen platforms as you can get. There are lifts, escalators and signs so again it is not difficult for a confident traveller.

Upon arrival at Narita airport, take the Narita Express to Tokyo. From there, you can take the Nagano shinkansen to Nagano.

I am planning to go there in Dec to see the snow monkeys in Yudanaka.

PS: JR pass is covered till Nagano. The train service from Nagano to Yudanaka is not covered by JR pass.

Use the websites recommended above to get a sense of timing, but I've found the easiest thing to do when touching down at Narita is to just go down to the JR office at the airport and tell them you want to go to Nagano. If you have a JR pass, you need to go there anyway to exchange your pass, and they'll reserve whatever seats are necessary and tell you which trains to get on. My Mom knows no Japanese, and their directions got her to Kyoto straight from the airport with no problems.