Is it possible to change the password for my own account using Set-MsolUserPassword if I do not have an administrative role? When I try that I get Access Denied error... Before calling that I call Connect-MsolService
No. The error you're getting, when trying to run the Set-MsolUserPassword cmdlet is expected. You have to have either the Password Administrator, User Management Administrator, or Global Administrator role assigned to you to be able to run that command. If you're looking to be able to reset just your own password, your administrator can go through the below article and set up self-service password reset for a select set of users or the entire tenant.
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