Does anyone know how I can rearrange additional categories at the product level?
This feature is not available. You can just short them by clicking on Name or Path.
BTW what you want to do by rearranging them?
I want to choose a more prioritized item from this list. This item should be first, but the sometimes prioritized item can be the last.
So one better solution will be to create a ContentReference type property (as we did in our one of the project) to select category (as allowed type) and use it. What you say?
I take it you want one additional category to prioritize other than the primary category?
If not you could just assign your preferred category as the primary category for the entry - moving the entry to the preferred category will change the primary category for you in that case.
I believe the order in that list is determined by SortOrder in NodeEntryRelation. While it cannot be arranged in the Ui, you can set it by code (using IRelationRepository)