Alpha 1.2.6 was the last version before Minecraft went to Beta, somewhere near the latter half of December 2010. Happily Mojang have included single player Alpha 1.2.6 in the official launcher. But sadly they have disabled multiplayer. So we need to use MultiMC launcher to get it to work...

Enter MultiMC

MultiMC is an open source Minecraft Launcher, which has many features, like managing multiple instances of Minecraft all to be installed at once with their own versions and configs.

You can download and install MultiMC Launcher from their website.

Once you have it installed and running, you need to add your Minecraft/Mojang account to the Launcher. Click the Profiles button in the top right and go to manage accounts. Click add account in the manage accounts window. Enter your real Minecraft player name and password here. MultiMC will ask Mojang if you're a paid up user and add your account if you are.

Close the account manager and click on new instance in the top left corner. Find the select vanilla Minecraft field and click the elipse [...] button. From the list scroll down to the alpha section and chose "a1.2.6". Click okay and let MultiMC download what it needs to.

Click once on the a1.2.6 icon and then on the "play offline" button. Let MultiMC download what it needs to and then when asked "Choose your offline mode player name" enter the name you asked to be whitelisted as.

Eventually you should see Minecraft start. Click on Multiplayer to go to the multiplayer screen and enter the server's address into the text box
119.252.189.29:26776

If you did the above all corectly you should be standing at spawn. However, you wont be able to move until you register a passcode.

First time you log in only: Hit the chat button (defaults to T) and enter
/register <passcode>
Where <passcode> is a short passcode you will remember for when you log in in the future.

Every login after the first: Hit the chat button (defaults to T) and enter
/login <passcode>
Where <passcode> is the short passcode you entered at the register command.

Remember to use the exact same offline player name and passcode each time or you will be kicked.