Dual booting with 2 hdds question

I have a new laptop in the mail, a clevo p375sm. I have windows already installed on the ssd. This laptop supports several hard drives, and I really want ubuntu on one of them for a dual boot. I want to install 13.04 onto the momentus xt i have in my current computer, then just plop that hdd into the empty drive slot on my new computer, then configure the bios to ask which to boot into. Do you think, given that the new computer has radically different hardware than the one I want to do the install on, there will be issues? Or should I just wait until the new computer arrives, install the xt, and do a clean install to it from the clevo.

I guess what I need to know is if the drivers will auto update or if there's a simple command I can enter to get the up to date ones, after installing ubuntu with a far inferior machine.