I just ordered a Mac that will be shipped with Mountain Lion. Is it possible to install Mountain Lion on my other Macs? Or do I have to buy an upgrade in the App Store?
No. The preinstalled copy of Mountain Lion is not licensed to be installed on additional machines.
From the Software License Agreement (2.A.):
Preinstalled and Single-Copy Apple Software License. Subject to the terms and conditions of this License, unless you obtained the Apple Software from the Mac App Store or under a volume license, maintenance or other written agreement from Apple, you are granted a limited, non-exclusive license to install, use and run one (1) copy of the Apple Software on a single Apple-branded computer at any one time. For example, these single-copy license terms apply to you if you obtained the Apple Software ￼￼￼preinstalled on Apple-branded hardware.
You will have to buy an upgrade in the App Store for your other Macs, however for non-commercial use you can install a single App Store purchase on multiple machines, given all the computers are in your control.