Google "dark detecting circuit" and you will find some other ways this can be done. Many use a photo transistor insted of a photocell. The circuit you posted and the one on the site you posted the link for don't look right to me (i'm not an expert though).
To hold the alarm on you could use a latching relay or possibly a 555 timer in monostable mode. (Google "555 timer" and you will get loads of info on them, or search this board.) It all really depends on your alarm buzzer and how much current (amps) it draws. A 555 can only handle 200mA. You show a 9v battery and a 12v buzzer. The buzzer might kinda work at 9v, might sound sick or might not work at all. Running everything at 12v would be better.
This is kind of a hooky alarm. I wouldn't trust it to protect anything valuble in real life, but it certinly could be fun to make.
The way they show holding the little mirrors up is awful crude. I would imagine they will slowly sag and misalign the beam and set off a false alarms.