List of categories sorted by Date?

I am developing a WordPress site for a client in which every month there are two posts published into a new category (named for the topic of that month).

I’m looking for a way to display the list of categories (topics) in chronological order. I’m thinking of two possible ways this can be achieved:

1. List the categories and order by date. I have not found a good way to do this, probably because categories aren’t time-stamped. Maybe there’s a way to look to see how recent the posts in the category are and use that information to determine the order the categories?

2. List the posts in a loop where I only display the category name for that post. The problem here is that since there are always two posts per category, the category name will show up twice. Is there a way to show only one post per category in a loop?

Thanks for your help in advance!