Yes, it is possible. We allow saving below fields from Prestashop to Odoo.
Note, that because we are saving/updating customer only when we download sales orders from Prestashop, below functionality will work (fields will be updated if they are changed on Presta) only with new downloaded sales order for this customer
Note that in Odoo standard there are no such fields on the Partner object in Odoo. So in our Prestashop connector we have create special custom fields for this that are added to the Partner object. See screenshot below
Also on the sales integration configuration, we allow to define fields mapping. Basically we can define here which information should be stored to what field. Note that it is done because you may use many available Odoo apps that can define those fields in other way. So in our connector we allowed to use not only fields defined by us, but your custom fields
You can check video below for overview of the functionality