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I understand you want to sync the current contacts and calendars to Office 365 with the desktop Outlook, and the current email account in your Outlook is not an Exchange account. If there is anything misunderstanding, please correct me.
Based on my experience, you can first connect the Office 365 account to your desktop Outlook, and then simply copy the contacts and calendars from the old account to Office 365 contacts and calendar folders. After that, the content should be able to synced to Outlook Web App(OWA) as well. Please refer to Connect Outlook to This Account to connect your Office 365 account.
You can also do this by exporting and importing PST files through Outlook. To do this, please follow these steps:
1. Export Outlook items to an Outlook Data File (.pst).
2. Connect Office 365 account to your Outlook.
3. Import Outlook items from an Outlook Data File (.pst)
You can repeat these steps for contacts and calendars separately.
Did the above reply answer your questions? Please let us know if you need any other help.