The instructions you linked to describe using NDISwrapper to run your wireless device from the Windows drivers. This is a legitimate way to get some wireless devices running on Linux, but if you can find a native Linux driver it's a better experience.
Wine is never needed to use NDISwrapper or to use any driver, but in the case of the instructions you linked to, Wine is being used just to extract a file from the installer .exe.
Do note that those instructions are old so may be out of date and/or there may be better solutions since that I'm not aware of.
You can download Ubuntu packages from another PC and transfer them, but it's an annoying and difficult process. You go to packages.ubuntu.com, find the right package for the right version of Ubuntu, and download the .deb file.
For example:
http://packages.ubuntu.com/xenial/amd64/wine/download
However the above link assumes you're using Ubuntu 16.04 (Xenial) AMD64.
Once you have copied the .deb file to your Ubuntu PC you install it with
sudo dpkg -i packagename.deb
sudo apt-get install -f
Where packagename.deb is the name of the file.
Installing packages like this is a pain because if it turns out that a package requires another package, you have to go and get those packages too, and so on.
In this case you'll need to get packages for wine, wine1.6, wine1.6-amd64, wine1.6-i386 and in turn those have a bunch more dependencies such as audio and video decoders.
If at all possible, it may be easier to run the setup.exe on your Windows PC instead and try and extract the resulting files from there.