Of course you do have to spin the wheel to get your prize... I did found the page I was talking about though!
It's
http://www.neopets.com/prehistoric/mono ... rize.phtml
If you spin the wheel and then check this page, you can see how many seconds there are left on your spin (drop the last three zero's to see the seconds). When you accidentially close the main monotony page, you can just stay refreshing this page... when the timer hits zero, it'll say "server=done" and although you won't see what prize you got, it'll be there, waiting for you in your inventory... (or if you've got hit by a rock for example, your pet will have lost some hitpoints...)
If you want a longer explenation, check pinkpt and go to "game secrets", you'll find it there!