My connection is really poor at home so I was wondering if I could download windows updates at the library where it is faster?
You can. However, Windows updates take a lot of time, and I don't know if your local library will like that. If you are using WiFi, plug in some Ethernet to your computer. WiFi is much slower than Ethernet. And the downloads only last for your current session, so if you put your computer to sleep or shut it off to bring it home to install, then your downloads will not be there when you get home and you have to re-download them. It is better to start up your computer one morning, let Windows update run in the background downloading updates, then it will alert you when it has finished downloading. Then update your computer overnight, so the updates install overnight.
Hopefully this helps.