In the address bar where the internet address of the school is, is there a little padlock just to the left of the name?
If you click on that padlock, box will drop down and there will be a box inside that says 'show certificate'.
click on that
Now you may have to look a the information it gives you there.
It's very likely that Safari doesn't trust that certificate for some reason so it might tell you that.
It's certainly the sort of thing that causes that behaviour.
The solution if that is the problem, is going into something called Keychain and telling it to accept the certificate as valid.
The keychain help is here but you may heve to dig around a bit.
When you say other websites don't work, do you mean you can't connect to anything? That is, Safari is being rejected everywhere?
If you above doesn't work if can get onto the net, you may be able to download another browser like Firefox and if you can get that to work then you can at least start homing in on what the problem is.
I don't think updates is the problem.
It might be useful for you to compare the two computers and see what is different one to another. You would probably do well to tell us what your network setting are (click on the wifi symbol in the bar top right and go to advanced) Compare these between the two computers too.
But most likely it is a certificate that's out of date or untrusted for some reason or the computer has been linked to a specific certificate something like that so start there.