No option to update WP in client’s backend

It does seems that installation was customized in some way, either in regard to user account or user interface.

If that is option at all I would politely inquire with last developer about customization performed. Beats digging through code.

As for core update specifically — the “manual” way would be to delete old core files and upload a new core version in exact place.

Of course make backups before you touch anything and be careful that you don’t loose any configs or user–generated files. If admin is customized it wouldn’t surprise me if the directory structure has tweaks as well.