Facebook has the option of setting up of email alert and also SMS alert. When you go to Account -> Account Settings, select Notifications, here you can set the notifications by giving the email id and also receive SMS alerts by providing your mobile no. SMS alerts are free of cost as you just receive messages.
From the screenshot below you can see for what items you can set the notification like someone sends you a message or someone adds you as a friend or someone posts on your wall etc.
Receiving an email alert is very simple as you just have to provide your email id as given below.
To setup an email alert, click the link ‘Send notifications to your phone via SMS’, when you click this link you get a window like the one below. Select the country and the network provider and next to give your mobile number. When you are done with these steps you will start getting the SMS alerts.