Some devices can access a desktop's internet connection via USB, but this is pretty uncommon these days. I think my old Droid Incredible could do that, and maybe (reaching back to the Middle Ages) my old Samsung Omnia. But I wouldn't be surprised if that feature is no longer available on modern devices, since wi-fi is so widespread.
Downloading .apk files is generally not a good habit, due to the risk of malware as well as the potential for encouraging piracy (as discussed above). I would still suggest purchasing a cheap router and setting up your own home wifi network. It would make your tech life a lot easier.
