I'm going to assume you are referring to a legal copy which you plan on playing on blizzard's official servers.
To cut a long story short, you will not need to download and install every patch. But you will have to install multiple patches. WoW bundles patches occasionaly so i'd assume at a minimum you're looking at 1-2 gigs.
I'm guessing it will be one huge patch upgrade from 1.0 -> 2.0 and then another from 2.0 -> 2.4 and then multiple small ones to whatever 2.4xx will bring you to the most recent patch (they often hotfix bugs in major patches).
Here's hoping you have The Burning Crusade OR a few friends who are playing with you. It can get a lilttle lonely otherwise