The question How do you describe an Asian person without using the word “Asian”? I’m actually a little insulted at the question and some if the answers provided by the other users as someone who is from Asia, but at the same time I’m amused by the question and the answers provided as well. 
I think @Avery_Moore should be clear as to exactly what Asian specific are you aiming for. What I’m trying to say is that Asia is a very large and populous continent which means there is going to be a lot of people who are going to represent Asia and a lot of countries that are going to differ depending on again what country specific you are trying to connect them with.
You can’t really describe a person from any continent and let it be the basis as to what the whole continent is supposed to look like. Not everyone from Asia has almond shaped eye, white skin, brown skin, dark skin, short height, good at martial arts (lols okay I’m just stretching the comment and being slightly sarcastic) etc.
Heck, I’m Asian, but I don’t have a name with Lee, Chin, Zhu, Sakura, Hotaro, etc. Which are popular and easy to place as people descending from Asia because when people generally think of Asia the top things they think are Japan, China, and Korea. True, yes, but there are other countries as well that are from Asia lols.
There are 48 countries in Asia, not counting countries that are dependencies or other territories. (Dependencies - a dependent or subordinate thing, especially a country or province controlled by another.)
48 countries in Asia (Asian as a whole)
China - Eastern Asia
India - Southern Asia
Indonesia - South-Eastern Asia
Pakistan - Southern Asia
Bangladesh - Southern Asia
Japan - Eastern Asia
Philippines - South-Eastern Asia
Viet Nam - South-Eastern Asia
Iran - Southern Asia
Turkey - Western Asia
Thailand - South-Eastern Asia
Myanmar - South-Eastern Asia
South Korea - Eastern Asia
Iraq - Western Asia
Afghanistan - Southern Asia
Saudi Arabia - Western Asia
Malaysia - South-Eastern Asia
Uzbekistan - Central Asia
Nepal - Southern Asia
Yemen - Western Asia
North Korea - Eastern Asia
Sri Lanka - Southern Asia
Syria - Western Asia
Kazakhstan - Central Asia
Cambodia - South-Eastern Asia
Azerbaijan - Western Asia
United Arab Emirates - Western Asia
Tajikistan - Central Asia
Israel - Western Asia
Jordan - Western Asia
Laos - South-Eastern Asia
Kyrgyzstan - Central Asia
Lebanon - Western Asia
Singapore - South-Eastern Asia
Turkmenistan - Central Asia
State of Palestine - Western Asia
Oman - Western Asia
Kuwait - Western Asia
Georgia - Western Asia
Mongolia - Eastern Asia
Armenia - Western Asia
Qatar - Western Asia
Bahrain - Western Asia
Timor-Leste - South-Eastern Asia
Cyprus - Western Asia
Bhutan - Southern Asia
Brunei - South-Eastern Asia
Maldives - Southern Asia
Dependencies or other territories in Asia
Taiwan – Dependency of china
Hong Kong – Dependency of china
Macao – Dependency of china
See, Asian people are diverse. Same with any other continent and country for that matter. Asian people can’t be put into one description lols.
What I would recommend is to focus on the country that you want the character to be from. It could not be by looks, but maybe you could insert their clothing, name usage, and mannerism. Those are the things you really can’t take away from any person who descends on that specific country.
People are mixed and matched now. And what I mean with that line is that a person could have a mixed heritage say a his/her/their mother is from a Europe country then we have father from an Asian country. Want to bet that baby isn’t going to have the cookie cutter Asian look? Yet still be Asian or European for that matter.
Rather than focusing solely on how to describe what they physically look like. Try incorporate details that would give away as to where that person originated from. Like I stated clothes, name usage, and mannerism are good to include. Not just looks, please. Okay, I said my peace on this matter and I hope as an Asian that it clears up the whole describe Asian person without using the word “Asian”?
So many countries, so many Asians…