Just to extrapolate, I do not really know what niche you’re serving and how big is the competition. The rules are a bit different for Craigslist type websites and for example paid listing websites.
My thinking is that because these listings are usually quite temporary (30 to 60 days max) that I should be setting the individual listings to noindex but I think the archive page should be indexed. Is this correct? If so, how would I accomplish this?
This is true if you have established website, the categories should be indexed and yes, if you have a lot of listings, you should not index individual posts. This is a bit tricky, because you can run into someone trying to sell something rare with long lead-in time. You want that indexed, don’t you? Or, if you do not have a lot of posts in a certain category, it might make sense to just keep posts there all indexed and let just 404 if they get deleted or expire.
The paid themes available might have some sort of logic implemented to set indexing status based on age, availability or other conditions. You can of course code it yourself or hire someone, TSF has documented API, and it is not an extremely difficult task.
It also depends on the niche. I have a feeling you read this article, where the author states lot of 404s is bad. If you are worried about 404s, Google wrote a decent explanation of why 404s are not bad and how to work with them, correctly. All the way back in 2011, yes, some SEOs parrot 404 is end of the world. It is absolutely not, especially with your type of website.
Feel free to follow up, but please with more details about your niche and preferably also URL.
Cheers,
Pierre