Tokyo to Shizuoka with JR

I am travelling with JR. East pass and I need to go to Shizuoka from Tokyo return. Can I use the pass or what otherwise is the cheapest option?

No, the JR East Pass cannot be used. Here are your options:
http://www.japan-guide.com/e/e6351.html

Thanks. What I am actually doing is travelling from Nagano to Shizuoka, and the route goes via Tokyo. If I do this trip and get off in Shizuoka I presume I have to pay part of the fare. Am I best off to buy my ticket for the Tokyo to Shizuoka part in Tokyo or when I get off in Shizuoka?

If time is not really an issue - then you can catch the Tokaido line from Tokyo to Shizuoka. It will take around 3 hours but should cost less than 3500 Yen. These trains leave every 15-20 minutes from Tokyo. I tend to also pay the extra 1500 yen for a Green Seat.

This option used to be quicker for me to get to Atami from Yokohama instead of catching the Shinkansen - but the Shinkansen will be the fastest by a long way (around 60-70 minutes from memory)

Use this website to find different train options....

http://www.jorudan.co.jp/english/

Have you noticed that JR East Pass does not cover TOKAIDO Shinkansen though it covers Nagano Shinkansen? So if you want to take Shinkansen from Tokyo to Shizuoka, you must pay the whole fee (ticket fare and the seat fee), which is around 6000 yen.

On the other hand the JR East Pass allows you to take ordinary Tokaido line as far as Atami. So if you take commuter trains from Tokyo, you will pay only 1130 yen (Atami to Shizuoka). But it takes about 3 hours from Tokyo to Shizuoka. Shinkansen will take you to Shizuoka in about 1 hour.

If you take non-shinkansen train, you can just pay Atami-Shizuoka fare at Shizuoka station. I do not know the exact system there, but if you show your JR East Pass to a station staff (inside the ticket gate), she/he will show you the way to pay it. If you use Tokaido Shinkansen from Tokyo to Shizuoka, you can not get through the entrance gate since you must have the Shinkansen ticket (reserved or non-reserved). In this case you must buy tickets (ticket fare and the seat fee) beforehand.