So I don't actually know how in depth of an answer you are looking for, but I will do what I can. This isn't really a "book smarts" question so much as a rote memorization question, as the Japanese use a historical era nomenclature that has little in common with the western systems and there is no way to "calculate" for era names.

38 AD would be in the middle of what is called the Yayoi period (弥生時代) in Japanese history. (running from the 4th century BCE to the 3rd century CE). It's considered very early history for Japan and the name actually refers to a kind of pottery. My Yayoi knowledge is a little sparse--as is my pottery knowledge to be honest--but if I am not mistaken Yayoi pottery was more technical than the previous Jomon period's (縄文時代) pottery. Jomon pottery was made using coiled rolls of clay and was relatively rough, where as Yayoi pottery was quite fine walled in comparison. So yeah, the period you are looking for is "Yayoi" and the name is similar to saying something along the lines of "the iron age" in English.

Just to expand the discussion a little, you may notice that I use "period" rather than "era" in the above. It's a conscious choice. The Japanese "Era" system doesn't start until a few centuries later. The Japanese era system is generally set by the Japanese emperor and is given a kanji (imported Chinese pictogram) name that is usually 2 characters in length although there are some outliers in the 8th century. Usually the era name is decided on the emperor's ascension to the throne although there are exceptions. The era name in the late 19th century was renamed from the Keiou（慶応）era to the Meiji（明治）era following the restoration of imperial power in the late 19th century. We actually entered Reiwa（令和） last year, so the traditional Japanese name for 2020 would be Reiwa 2.

The names are pretty commonly used by everyday people even now. You use them on many official documents (although plenty of places accept western numbering styles as well). For example, when I go to city hall to renew my foreign residence card I need to write my birth year as Heisei 6 (平成６年) also known as 1994. Japanese kids learn Japanese history in the context of eras as well. The words are also used in discussions of culture "That hairstyle is so Showa," "I really like the culture of Taisyo romance." I think you get the picture. Generally names refer to something or express some desired guiding principle for the country. There are some hiccups though, the current Reiwa era period received some push back because the modern common interpretation of the first character is literally "Order," "Law" or "Command," while the second one translates to "Peace" or "Harmony" (令 Law 和 Peace); some of my left leaning friends here saw this as a dark allusion to authoritarianism. I can't speak to whether they are right or not, but my assumption is that the intention was to evoke a more esoteric reading of the character, which comes across as "Splendid," "Beautiful" or "Awesome."

For further readings check out just about any introduction book to Japanese history.My professors used the Cambridge History of Japan for general overview during my undergrad.