If you are looking for a "No-CD patch" because you want to play Quake 3 on a modern 4K monitor with high refresh rates, a source port is a much better option than a simple patch. Since id Software released the source code for Quake 3 years ago, the community has built improved engines.
. During installation, ensure you select a "Full Install" if prompted, as a minimal install may still look for the CD. Step 2: Update to 1.32c quake 3 arena no cd patch link