The Last Outpost's Minecraft server runs the latest version of Minecraft, so be sure you're updated to the latest version before attempting to connect!
Firstly, ensure you have a premium account! This server is not cracked, you must have a premium account to connect.
Secondly, make sure the login servers are online. If you're having issues connecting, visit http://xpaw.ru/mcstatus/ to see if Minecraft is down, if it is, wait it out, if it isn't, consult Jaysee, or Andrewthetechie. If you don't do this, you're only wasting the time they could be using for more serious issues.
Lastly, connect! The IP is "mc.last-outpost.net." (minus the speechmarks). Now, you can connect to the server.