So if you had access to the gaming site and this access was blocked just after this script (->pop-up) came up, you may be right. But this is definitively a strange way to limit asccess to "special" sites. If I want somebody not accessing a site, he should never access it. May it be that a site contains illegal or not approbiate material or for another reason. If I want to protect my network form viruses, it will be to late to "kick" somebody off a dangerous site after one hour, because a virus might have already been installed.
So all network protection should not run on the client (->your laptop), but on the server providing internet access.
Is it posible that your school runs a kind of fair use policy, which allows "nearby" unrestriced internet access in the pupil's freetime, but limts the access during the school hours. That would make sense. I do not think it is a good idea trieing to access sites that you should not. In fact I think you school has a policy for that and you may get into trouble, if you do not respect it. There are often ways to circumvent limitations, but if you get caught, you can not even say that you did this by accident. Due to that fact that all network traffic runs through their servers they can see where their pupils are surfing to. So it will be better to do "private" stuff on your private laptop with your private internet access, if your school does not give permission to use their equipment privately.