Yes, you can just say "Sync Lead to SFDC: Auto Assignment Rules" and it will update the Contact's information. The issue you were having is that you were previously attempting to effectively change owner or assign the Contact to a Queue, which automatically creates a Lead instead.
The process to use is to have 3 sync flows - one for SFDC Leads, one for SFDC Contacts, and one for Marketo only leads. That was I can assign a new lead to a Queue, an updated Lead to an Owner (or whatever), and add a Task to a Contact.