I'm happy to be corrected, but I don't think you can get into the BIOS without a CPU. However (if you had the flashback feature) then you can update the BIOS that way without a CPU.
To run a 13th gen CPU you will need BIOS 2004 or later (or the equivalents for your board). So, I think it's true to say that if the BIOS is older than 2004, then you will need a 12th gen CPU to access the BIOS and update it that way.
Z690 Hero, 12900K, BIOS 4301, MEI 2512.7.3.0, ME Firmware 16.1.35.2557, 7000X Case, RM1000x PSU, ASUS TUF OC 3090TI, 2 x 16GB Corsair RAM @ 5200MHz, Windows 11 Pro 23H2, Corsair H150i Elite AIO, 4x Corsair RGB fans, 3x M.2 NVME drives, 2x SATA SSDs, 2x SATA HDs.