Let $url be the URL with the current version of UnRAR, and do the following to overwrite the current source with the new source:
wget -O unrar.tar.gz $url rm -r third_party/unrar/src/ tar xf unrar.tar.gz mv unrar/ third_party/unrar/src rm unrar.tar.gz
Commit these changes and upload a patchset. This will make it possible to generate the patch file later, and make review easier. Let $raw_src_hash be the resulting git hash.
Now apply the Chromium patch file. This command will apply the hunks that apply, and create .rej files in //third_party/unrar/src for hunks that do not:
git apply --reject third_party/unrar/patches/chromium_changes.patch
For each .rej file, identify whether it still applies and manually fix the conflict, then delete the .rej file. Then regenerate the chromium_changes.patch file with:
git diff $raw_src_hash -- third_party/unrar/src/* > third_party/unrar/patches/chromium_changes.patch
Update README.chromium to mention the new version and source URL and upload a CL. You did it!