Yes, when you run the updates the packages will be stored in /var/cache/apt/archives. I'm not sure if these will be deleted when you restart although it seems likely.
Another thing you can do is run apt-get -d updates from the terminal. This will download the updates but will not update your system. (I've never used this before, and since my system is up to date, I have no way of verifying. I'm going off the man pages here, so I would be grateful if anyone can either correct me if I'm wrong or verify if I am.)
You can now copy the updates to all your other systems, and run apt-get install in all of them. This will install the updates.