Customer External ID can be exposed on the customer list when you edit / customize the customer list view. As of the moment, there is no way to expose this field on the customer record itself. This is covered by Enhancement 112229 - Add a "Show External ID" preference to the UI. The External ID field cannot be manually edited. This is not also available for Inline Editing. The only way to update or change customer external id is via CSV import update.
Here is how you update the external id on customer records via CSV Import Update
1. Navigate to Setup> Import/Export> Import CSV Records
2. Set Import Type to Relationships
3. Set Record Type to Customers Only
4. Set Character Encoding to Western (Windows 1252). Note: Choose another character encoding format if you use an international or Macintosh version of Microsoft Excel, or if you typically use special characters.
5. Set CSV Files to One File to upload
6. Hit Select… button then browse and select the saved CSV file to be uploaded.
7. Hit Next.
8. Set Data Handling to Update
9. Hit Next.
10. On the Field Mapping page, make sure to match External ID and Internal ID of your CSV file to the appropriate NetSuite fields. Note: ID is mapped by default to Internal ID. Make sure that ID is not Customer ID but the system generated internal id.
11. Hit Next.
12. Type in the Import Map Name
13. Hit Save & Run
Note: The confirmation message contains a link to view the import job status. Here you will see if the records were imported/updated successfully and if not, you will see here the cause of the failed import/update.