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.
Shaelux - Warlock Lvl 18
Shaelur - Paladin Lvl 22 Currently in Walkthrough: viewtopic.php?f=3&t=2746&p=31373#p31373
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.