There are fifteen* in-game books dealing with or somehow discussing Morian Zenas and his journey to Oblivion, http://www.imperial-library.info/content/doors-oblivion and the http://www.imperial-library.info/content/racial-motifs series. I always found the character of Seif-ij Hidja interesting and so I devised a Skyrim play-through based on him, long before Elder Scrolls Online had been announced.
As such, I based what few backstory details I had on the information held in The Doors of Oblivion. They led me to give an approximate time for Morian's journey as between 3E410 and 3E432. Due to the book already being around in 3E433 (year of the Oblivion Crisis) it had to be before then. The main clue that led me to set 3E410 as an earliest date is
Jagar Tharn's time is established as being 'some years ago'; if we assume 'Jagar Tharn's time' is referring to the time he spent posing as the Emperor, which I believe it is as Battlespire was destroyed by Mehrunes Dagon as part of Jagar Tharn's consolidation of power, that this 'some years ago' had to be between 3E389 and 3E399. Add on enough time for it to become 'some years ago' and you get to about 3E410. There is leeway for it to be a little earlier, but even with a generous interpretation of 'some years' we have an absolute earliest of 3E390s. There are very few further clues to narrow it down further - I suppose you could make an argument that Divayth Fyr must still be alive and the player does have the possibility to kill him during Morrowind, so you could place an upper limit of 3E426 - but for the purpose of this discussion it is the earliest possible date that is most important.
So, all well and good - it has to be 3E390s at the earliest. BUT ... Racial Motifs completely contradicts this. During the course of the books, Alfidia Lupus references Morian Zenas and Seif-ij Hidja, as well as Divayth Fyr, so there is no real possibility of them discussing some other Professor Zenas. It culminates in Morian Zenas going to Oblivion:
Convinced it's the same journey? Here's the problem:
Alfidia Lupus was alive in the second era.
The fore-word of Racial Motifs 1 only establishes:
You could make the argument that just because those are the dates she was Imperial Ethnographer, it doesn't mean she was researching and writing her series during those dates. That's probably correct, the termination of that position was more likely due to the death of Savirien Chorak than her leaving, which the books http://www.imperial-library.info/content/brothers-darkness and http://www.imperial-library.info/content/pocket-guide-empire-third-edition-all-eras-man-comprehensive-history-our-historyestablish as 2E430, contradicted by http://www.imperial-library.info/content/pocket-guide-empire-first-edition-wild-regions which gives it as 2E431. So yes, there is nothing to suggest it was precisely 2E430/1.
However, you would need approximately 870 years between 2E430/1 and the earliest time that The Doors of Oblivion suggests for Morian Zenas's journey. Unless Alfidia Lupus was an Altmer (unlikely, very few Altmer have been shown to have family names and she needed to search out Morian Zenas for information about the Altmer motifs**) it is unlikely she would live this long, meaning that Morian Zenas's journey, according to the Racial Motifs series, must have been closer to 2E400s. Besides that, Racial Motifs 4 states that Morian Zenas was around at the founding of the Arcane University:
And it is known from http://www.uesp.net/wiki/Oblivion:Hannibal_Traven that the Arcane University was founded in the second era:
So the known facts don't even suggest an extended life for Alfidia Lupus is the explanation for the discrepancy.
Leading us back to the original question: when did Morian Zenas go to Oblivion? There's a choice between some time in the second era, most likely 2E400s, or 3E390s at the earliest ... depending upon which source you side with. And no real information to account for the discrepancy. The sole reason I'd probably argue for the latter over the former is, if Morian Zenas and Seif-ij Hidja had been writing in the second era, it's unlikely that their books would still be widely available in the third era, the best part of a millennium later.
* I'm discounting http://www.imperial-library.info/content/oblivion - whilst it is the book written by Morian Zenas, it gives no real information or clues that allow the reader to theorise upon its historical context and from there determine an approximate date. It does establish an earliest time of late first era, by referencing the Alessian Doctrines (known from http://www.imperial-library.info/content/last-king-ayleids to be 1E361 or thereabouts) and a further work
which combined give us an approximate earliest date of 1E1360. However, as this date easily accommodates either timeline, it is in no way helpful.
** Pure speculation, but for this reason I believe Alfidia Lupus to be Imperial. It is known from Racial Motifs 10 that her home town is Skingrad, but that in itself does not mean she is personally Imperial. The reason I believe that is it is the only extant race for which she did not reference a native source, implying she doesn't need one because she is one.