You have a lot going on here!!
I really cannot be sure on this. It's POSSIBLE that there is a misspelling or abbreviation, but I cannot be sure. And it would depend on what papers you have and from where they were. If the papers you have are American, then millions of mistakes could have been made since Russia
Mich- MIGHT BE short for Mikhail or Michael. No promises on that. But if that were the case, Alexander could be Mikhailovich/Michaelovich (son of Michael). Micha (said Mee-h'a OR Mee-sha) is a nickname for Michael whereas Misha (said Mee-sha) is a nickname for Mikhail.
A patronymic name is not usually a surname in Russian culture. For example: If Alexander is the son of Peter (Piotr) Michwich, then his name would be Alexander Piotrovich Michwich. If he's the son of Michael Smith, then his name would be Alexander Michaelovich Smith. Piotrovich and Michaelovich are male patronymic names. They honor the father. Female patronymic names are a little different.
Not knowing if Michwich is a misspelled patronymic name is a disadvantage here. If you knew Alexander's father's name, it might help. If Dad is Michael Michwich, then we know that Michwich is a surname and not a misspelled patronymic name.
So, taking all of this into consideration ... just a stab in the dark, and a guess at best: Michwich COULD BE Michovich/Michowich/Michowicz/Michowitz (son of Mike). The spelling could change based on his region, religion, ethnicity, port of departure, ... IE: usually ... wicz = Belorusian, witz = German or Yiddish/Jewish, vich = Russian ...etc. But there are no guarantees. wicz is seen in Poland, Germany, ...
You might have other clues that "tell" the family's story "hiding" somewhere and not even know it. If he was Jewish, he probably brought a Menorah to America, for example. Maybe pictures ...
If YOU are not Jewish, and no one in your family is Jewish, do not assume that he might have been. Because most languages, including English, did not have spelling rules before the 20th Century, someone named Michaelowich could easily change the spelling to Mikhailovich in order to "fit in" to the area. Spelling errors can occur in so many places in Immigration ... ships' logs, port of departure records, port of arrival records, Ellis Island, etc.
And since the Russian (Cyrillic) Alphabet is nothing like the Western Alphabet that is used in America and most of Europe, there is all kinds of room for error. As a quick example, the Russian letter B /veh/ sounds like the German letter W /veh/ which is represented in English as V /vee/. So at port in Hamburg Germany someone hearing Mikhailovich might write Michaelowitz. And at Ellis Island, another person hearing Mikhailovich might write Mikaylovicz.
So sorry, but you have a lot of research to do.