A much cheaper and probably better option than a private guide is the audio guide.
For 4 euros you can hire a small audio device that plays commentaries as you go around.
There are about 40 numbered points at the Alhambra and when you press the number on your MP3 player device you can hear a commentary.
The production quality is high and it is just as good if not better than a real human.
You can get them from the information desk next to the ticket windows.
You have to leave your passport as security.
More info herehttp://granadainfo.com/alhambra/audioguides.htm
If you are a real cheapskate you can eavesdrop on other tour guides.
I do this occasionaly, some of the guides are good and some are very bad.
If you really want a guided tour Click here
There is an excellent Virtual Walking tour of the Alhambra. Click here