I assume you have a Mini without a CD/DVD drive... or what?
Which Mini is it & what OSX do you want to install?
If you don't know the model, find the Serial# & use it on one of these sites, but don't post the Serial# here...
How to find the serial number of your Apple hardware product...
Mac mini or iPad mini?
There is a BIG difference?
If iPad mini, then yes, you can use iTunes on the Mac to upgrade the iPad mini iOS to 7.0.4
If Mac mini, then you can maybe put the Mac mini is "Target Mode", connect a Firewire cable between the Mac mini and the iMac (adapters maybe needed depending on what ports are available on each system), then mount the Mac mini's Target Mode disk on the iMac as an external disk, and apply the operating system update the Mac mini's Target Mode disk.
Again for a Mac mini: Or on the iMac, enable System Preferences -> Sharing -> DVD or CD Sharing, then the Mac mini should see the iMac's DVD drive as a networked device on the Finder side-bar. In theory you should then be able to use the iMac's DVD to load your software on the Mac mini.
As I do not have your exact setup and operating system flavors, I cannot be sure everything will work as explained. (that is to say "Your Mileage May Vary").