It's less of a hassle to update the BIOS before putting the new CPU in, specially if you're selling the old one. if the CPU is not compatible with the BIOS version you're on currently (which it must be if you haven't updated the BIOS in a while) you wont be able to update the BIOS with the new CPU installed and you should switch back to the previous CPU (or other supported CPU), update it, and then put the new CPU.
Its also better to stay on the latest BIOS version, whether you're upgrading the CPU or not. (there no reason to stay on an outdated BIOS version unless for specific reasons like bugs, crashes [due to new BIOS] , etc.)
FYI;
- Old BIOS versions - Supports older CPUs, might not support the latest.
- Latest BIOS version - Supports both old and new CPUs.
TL;DR : Update the BIOS with the CPU you currently have, before installing the new CPU.