You can buy an external enclosure for the old drive such as this one from macsales.com that I bought: http://eshop.macsales.com/item/NewerTech/FWU3ES2HDK/
As to the new drive ... unless the old drive is a 1 TB drive and the new one is 120 GB, I would use the newer and presumably more reliable drive in the new MBP.
As to using the TimeMachine backup to restore, you would still be re-installing the same and older-hardware device drivers.
You could use the older drive after you transfer your files to re-install the new OS (Mountain Lion is due out) using Command-R and the internet link to get a new system-tuned OS. Or you could use CarbonCopyClone or SupderDuper! to use the old drive as an external bootable clone that can be used to reboot your entire system when/if the internal disk or the OSX on the internal drive is is trouble.