Kamakura: Faster and cheaper way to reach

About Kamakura: what is the faster and cheaper way to reach it from Tokyo? Except train I mean.
Can I use Suica Card anyway?

Fastest = train. Cheapest = walk. Suica = yes.

OP said except train.
It should be:
Fastest = car. Cheapest = walk. Suica = yes.

You can't reach Kamakura without catching a train. Car will be expensive, either rent-a-car or taxi.
However, local trains to Kamakura are inexpensive . It depends where you are departing from (Tokyo is a huge city!) but for example from Shinjuku it costs 920 yen (one way) and you may use your suica on them.

Walking isn't a terrible idea if you are a fit hiker. It will take you about 10 hours from Shinbashi to Kamakura on the Tokaido road and there's plenty of great spots on the way. Of course you won't see anything in Kamakura by the time you arrive and can't go back the same day, but that would be gloriously free.

From Shinjuku 920 yen train or bus?

Your options are explained in this article; they all include one or a few train rides.
- japan-guide.com: How to get to Kamakura
http://www.japan-guide.com/e/e3115.html

Generally a cheap transportation mode tends to be not fast.
In this article, it is explained that Odakyu's Enoshima Kamakura Free Pass makes the cheapest way, but a trip to Kamakura which this pass covers takes more time than that by JR.

Why no train?