There are different ways to play minecraft beta/alpha, classic and pre-classic. I'll list them all for you.
This is really the worst way, don't use it to play in these versions... It may be good for discovering them, but to fully enjoy them, it's not worth it. I swear, don't do it.
You can do anything (including install mods), but it's just not convenient. There are better alternatives.
My favorite launcher, it “simulates” the vibe of the old launcher. It also offers lots of extra things.
It's personally my favorite launcher. Check it out here : [- DOWNLOAD -]
Classicube is a great alternative to discover Minecraft Classic (especially in multiplayer mode).
I can't tell you much more about it, as I tried it out some time ago... But I strongly advise you to give it a try!
[- DOWNLOAD -]
This launcher is very different from the others in that you can play recent versions of the game just as well. But there's a lot you can do with it.
This is probably the best all-in-one launcher for enjoying Minecraft (modded, etc.). It lets you create Instances (lots of different .minecraft to separate your modded minecraft from your vanilla minecraft, for example).
So you can enjoy Minecraft Beta and alpha with it! (much less interesting for preclassic and classic, but it's possible!)
[- DOWNLOAD -]
I found this on internet because I wanted to play it without jailbreaking my xbox360 (I should do it one day), it's not optimal but if you want to rediscover this version without bothering it's a good alternative! There must be a new, more optimized version, so don't hesitate to let me know.
Several ways to play Minecraft on a Web browser.