How do I "buzz" another skype user?

You probably can't "wake up" a computer without using something like LogMeIn.

All the called party has to do is leave their Skype/computer logged in and in stand-by mode so it can receive a call at any time. The audio sound of a telephone ringing should be sufficient to alert them that you are online and they should answer the Skype call.

There are many Skype tutorial videos on youtube if your other party needs something visual to look at to learn how to set up in stand by mode to receive calls.