Where exactly have you bought the upgrade? In our webshop this is only possible if you enter the serial number of your existing Melodyne product. Since you didn't have one, I suppose you bought it from somewhere else?
Anyway, to answer your question: yes, if you purchase Melodyne essential, you will be able to make use of the Melodyne assistant upgrade.
I want to record my voice (only) singing a song, download it to Melodyne Assistant, and be able to correct pitch, time, etc. Is the the correct product for what I'm trying to accomplish?
This is possible in all Melodyne products. However, Melodyne essential gives you very basic pitch and time editing tools only, while Melodyne assistant also comes with tools for pitch modulation and pitch drift, for example, which allow for more detailed editing. See also our comparison chart
You could as well start with Melodyne essential, and upgrade to Melodyne assistant only later, if you find that the Main Tool in Melodyne essential is not sufficient for you.