I'm going to assume that by "Asian" you mean it in the more common American sense of "East Asian", and not Desi, since you're referencing the gold rust era. Not that that makes this an easier question to answer.

The 14th Amendment of the constitution says that "All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside." This gets interpreted as, you're born there then you're a citizen, and that's why you get phenomena like anchor babies.

The problem was, this hasn't always been applied, or interpreted in the way that we tend to interpret it today. The Chinese Exclusion Acts barred Chinese from becoming citizens and heavily restricted their ability to enter the country, whether as migrant workers of immigrants. This became a major court case, going all the way to the Supreme Court, when an American-born Chinese named Wong Kim Ark was denied entry, even though he had been born in San Fransisco, his parents had been there legally, and by all counts he should have been given US citizenship. In the end, the court ruled in Wong's favour and his status as citizen was recognised. This was in 1898.

Note also that First Nations people were granted full citizenship based on an 1884 law. So for many people, being born in the country, no matter how far back your family's presence goes, wasn't always sufficient.

Other cases happened later on arguing for the rights of citizenship. in 1921, Easurk Emsen Charr (Cha Ui-sok), a Korean-born US Army veteran, argued for his citizenship based on his military service. The Federal District Court of Missouri ruled against him because of was "of the Mongol race", and that was the end of it for him and those in his position until many of the Exclusion Acts were repealed, notably in this case the McCarran-Walter Act of 1952. Charr wrote an autobiography titled The Golden Mountain: The Autobiography of a Korean Immigrant, 1895-1960 in which he wrote about this period.

Still, the earliest naturalised US citizen that I'm aware of would be John Huston, a Chinese-born American who came as a child in 1829 to New York. The situation was better in the eastern states. Much of the exclusionary policies adopted by the federal government were originating from and largely pushed by California. In New York, things were different. Two other Chinese Americans are also recorded around that time, also living in New York, named John Islee and Lesing Newman. You also had Chang and Eng Bunker, the conjoined Chinese-Siamese twins about whom the term "Siamese twins" was popularised, who became citizens in 1839.

I couldn't tell you exactly who was the first, but if not someone just listed, it would have been someone in a similar situation and time period as those above who we just don't have records of.

There were of course East and Southeast Asians in what is now the United States before then, but not in places that were the United States at the time, or else denied citizenship. For example when Hawai'i became a state, a large portion of the population was Japanese, but they were not given citizenship either at the time nor any time before.

If you're interested in this topic, I can't recommend Erika Lee's books enough. Much of the above is based on The Making of Asian America: A History and At America’s Gates: Chinese Immigration During the Exclusion Era, 1882-1943. They're both very well researched and she's been generous with the number of photos she's included of those involved.

tl;dr: Your gold rush dates aren't too far off