There are many post words in English, including not just as nouns and verbs but also more exotic parts of speech such as adverbs. There are also post- combining forms (like prefixes), which is probably what you were thinking here.
But it’s not.
In this case, post is neither a noun, a verb, nor an adverb, nor is it a prefix, either. Rather, it here functions a preposition whose object is ingestion.
And we never separate prepositional objects from their governing prepositions with hyphens. So what is written there is just fine. It’s just a preposition equivalent to after in your example:
Samples were collected one month post ingestion.
This is a comparatively recent revival of the old Latin preposition post, which gave birth to the prefix post-. Oxford Dictionaries Online says this freestanding practice began in the 1960s. Me, I would not be surprised if that could be antedated — or by quixotic extension, dated ante the 1960s. :)
You could have a post-ingestion problem, or you could have a problem post ingestion.
Ad-hoc compounds formed with post- are almost always written with the hyphen, but it’s completely a matter of house style, with some historical practices (read, habits) weighing in. Check with your publication’s style guide to be sure.
The only exception is that you must use the hyphen when the form you’re combining with takes a capital, so post-Pliocene or post-Columbian compared with an established postmillennial or postnatal.