No, you can't run Zimbra under cygwin.
Have you heard of a company called VMware? They produce virtualization software, you install it on your machine (Windows in your case) and it allows you to install another operating system on the same machine and run it as a virtual machine. So, you install VMware Server, you then install a version of Linux to run under VMware and then you install Zimbra on that version of Linux. It appears to your LAN just like another server running on it's own hardware with a LAN IP address. As I mentioned, to do this you would need additional RAM on your current Windows server.
Using VMware isn't an ideal solution but you certainly can do it. Have a look at the VMware site
here and the free server software
here.
There is already a Zimbra Appliance that can be installed on VMware, have a look on
this page. The appliance only needs to be downloaded then run under VMware, you don't need to get a linux distribution working as it's already in the Appliance.
Does that help?