If you are using something like the Optibay drive from MCE Tech, you an put either drive in the caddy. I prefer to make a single change at a time, so when I did this I put the new drive in the caddy and left the old working boot drive alone.
As for installing from a USB thumb drive (I assume this is what you meant since you weren't clear), you shouldn't have to. You can download the Mountain Lion installer and run the app. During the install process you can choose what drive to install the OS on, simple choose the newly installed and formatted SSD.
Once you have both OS installed, the system will remember the last drive that was selected in the Startup Disk preferences and store that choice in NVRAM so that it will continue to start from the preferred drive. You can hold the option key at boot to prevent this choice and pick from all bootable drives when you want to switch things on a specific boot.
Most people find that having two OS internally is confusing. Spotlight will try to find the correct version of Safari if you don't keep them both updated, but sometimes you end up running the wrong version of an app. This is confusing at best, and troublesome most of the time and can cause data issues when the wrong version of an app opens data it didn't expect from a newer version of iPhoto for instance.