Looking at the integration guide from HubSpot, you can sync categories from Outlook only if you have an Operations Hub Starter subscription or higher. If you don't have access to those custom field mappings, you won't be able to automatically sync this data.
Your other option is exporting your Outlook contacts, along with their category, and manually importing that list into HubSpot to update records in bulk. This option doesn't require any additional HubSpot subscriptions, but it will be a manual process and contacts will be updated only when you run the import, not in real-time.
Looking at the integration guide from HubSpot, you can sync categories from Outlook only if you have an Operations Hub Starter subscription or higher. If you don't have access to those custom field mappings, you won't be able to automatically sync this data.
Your other option is exporting your Outlook contacts, along with their category, and manually importing that list into HubSpot to update records in bulk. This option doesn't require any additional HubSpot subscriptions, but it will be a manual process and contacts will be updated only when you run the import, not in real-time.
I've ended up using Outlook Sync because you can't export contacts from Outlook Mac or 365 in browser as a CSV file. So sync was my workaround. Do you know of a way to export from Outlook at a CSV?