Can I download updates for another computer? I have a Mac Pro that needs a lot of updates, but my home internet is very slow. I have a laptop, however, with access to better internet. 

Is it possible for me to access the updates on my laptop, download them, and then move them over to my Mac Pro and install them? If so, how?
A lot of software updates (particularly OS, driver and firmware updates) can be found on Apple's Support Downloads page. You can download them like any other file and just transfer them to your Mac Pro at your convenience.

For software on the Mac App Store (including bundled apps like iPhoto), there aren't separate downloads. You could try downloading the full apps on your MacBook then copying the application bundle over.

For third party software, there are often separate patches available for download from the developer's site.