Yes, changing the Ticket Field Name would automatically change the Label for existing tickets and the Field would also be automatically updated. Please navigate to Admin > Workflows > Ticket Fields click on the field to be taken to its properties window where this change can be made.
However, in the case of drop-down fields, if one of the choices within the drop-down is deleted or edited, the value would not be updated on existing tickets. The value of the field, in such cases, would revert to Null(...).