No the memory unit cannot be plugged in because you cannot physically remove the memory unit. If you own the arcade one, the 256mb hard drive space is built into the console as far as im aware but like others have said, you can transfer the data over to your new console or save it to a hard drive like I mentioned earlier. That would be the best thing to do in my opinion, put all the data on like a 1gb USB stick or something and then put it on your new one.
Game demos are usually about 800mb - 1.2gb so it would be better having the 250gb console if you want to download them. You do get a lot of good game demos on xbox live e.g. Fifa, Forza, Battlefield there are hundreds. Also, you can download games from the Xbox Live Arcade which you can store on the hard drive.
You can save your games onto the hard drive and it does help with loading time on some games but in my opinion it's always been negligible. I tried it on Fable 3 as there is so much loading in that game but didn't find it helped that much.
One good thing though, I damaged my Fifa 10 disk last year, so I borrowed a copy off my friend, saved it to the hard drive, then just used my copy of Fifa 10 to certify the game, then it plays it off the hard drive. Saved me having to buy a new copy! So it is useful for backing up games in that respect
Hope that helps!