There are early inscriptions that show confusion between oe and ē. For example ceperint (CIL ii. 1964. 4. 27) is datable to the late first century AD (it occurs alongside coeperunt in the same inscription). There are more examples, like amenus, citaredus, phebus from Pompeii which probably means they belong to the first century too.
This diphthong is hard to get a handle on from inscriptions because instances of it in the Latin language are so rare. Most instances of Old Latin /oi/ develop to /ū/. And only a fraction of inscriptions can be securely dated at all to a specific century — which means there are a lot of monophthongal spellings sitting around whose age we're basically just guessing at.
This is going to be way more than you asked for, but a simple definite answer to the question of what the earliest inscription with an e-spelling is, is a bit hard to give. And even if we could do that, it wouldn't tell the whole story.
If you're looking for the earliest evidence that some speakers were pronouncing a monophthong for oe, the first century CE is probably what you're after.
Sometimes handbooks will tell you, on the strength of such evidence, that the change was accomplished in the 1st century AD, but the impression this may give can be misleading. Some speakers had a monophthong that early, sure. But as with the monophthongization of ae, there was probably considerable lag (on the order of a century or more) between the existence of speakers who had the change and its diffusion throughout the empire (and its eventual acceptance by the grammarians).
How long did it take for everyone to adopt the monophthong? That we can't really say. Well over a century after the first inscriptional evidence of the monophthong we still have a grammarian, Terentius Scaurus, attesting clearly to a diphthongal pronunciation. (Note: he also seems to describe a diphthongal pronunciation of ae, even though spellings of this diphthong with E were already profuse in his day. Even Marius Victorinus in the 4th century appears on the face of it to describe a diphthongal value both for ae and oe.)
Occasional spellings (and they are rather rare, which you'd expect given the rarity of the grapheme) with e tell us only that some people had a monophthong. But given that the change is universal in Romance, it's hard to imagine that its adoption across the social scale was all that late. By the third or fourth century, monopthongal pronunciations must have been commonplace. The evidence of Marius Victorinus seems to suggest that there were even then still diphthongal holdouts, but they cannot have had any influence on the development of the language in ensuing centuries (and it's possible Victorinus is really talking about spelling rather than pronunciation, just in a very weird way.)