Maybe check blizzards website, they have free trial on the retail version at least. Maybe there is something similar for classic since it's through the same subscription.
Edit: And no you can't play this game without downloading it, even if you only want to "try" it.
Ten Storms EU Alliance
Finnigan - Dwarf Hunter Lvl 30
Portalmaster - Gnome Mage Lvl 19
Shaelur - Dwarf Paladin Lvl 22
Shaelus - Night Elf Druid Lvl 50
Either sub for a single month, it's only 15 bucks, or try the game out on a private server. Other than that, you don't have many options.
Temaius - Paladin