You're first question is pretty easy, you use the patch command. The way I do it is to jump into the directory containing the source code to be patched and then run this:
patch -p1 < /path/to/patchfile
You should see the patch applied along with any errors that happened.
Now your second question is potentially huge. There are a ton of kernel compiling howtos out there (including some excellent ones in the Slackware forum) but if you would like to be more specific about what distro you have and why you want to upgrade, maybe we can give better advice.