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.
In a comment, I wrote:
Because you should only use hyphens when some confusion would otherwise arise, which misparsing are you attempting to avoid here? Books don't hyphenate this.
In a comment, FumbleFingers wrote:
Yeah - hyphenating just still-fringe doesn't look too bad either (but just then-still looks weird). As a matter of fact though, when I just searched Google Books for the structurally-identical sequence the then still unknown (person, thing, idea) I didn't see any hyphenated instances in the first couple of pages of 636 results returned.
Best Answer
In some cases of dangling hyphens, the hyphen is put (back) into a word that used to commonly have it or occasionally still does. For example "inter- and intra-species". You may be able to do something similar in this case: "neuro-degenerative and -inflammatory".
If you couldn't get away with the hyphenated forms, this wouldn't work. A good internal peer review is of course necessary in grant proposals - hopefully you have someone who would nitpick things like this.