Welcome to the forums
You will certainly require a MX record for things to work. Until then you could setup a Split DNS - Zimbra :: Wiki architecture which will work just fine. This is due to it being sat behind you ISA on a private LAN.
Also, if you post these details it will help us even more
On another note I recognise that school name ... what region are you based ?
su - zimbra
dig _yourdomainname_ mx
dig _yourdomainname_ ns
host `hostname` <- note backticks and not double quotes