Aleksandr Patapovich Kvashuk - born around 1879 in Podolsk, Ukraine

So with the birthplace "Podolsk", I can only find the place Kamjanez-Podilskyj,Кам'янець-Подільський, Oblast Khmelnytskyj, Ukraine.

I also checked the "family name" once and find only ONE Kvashuk near Kishinev. The place is now in Moldova. There is also a patronymic Artemonov (son of Artemon) in the entry, which brings me to the next topic. If he was Russian/Ukrainian, then you won't get anywhere with the name Kvashuk, because fixed family names were only introduced after the Soviets.
The remaining Kvashuk are all shown for USA, so will be a self-chosen family name after emigration. My theory.
Otherwise, the search query on Ancestry with the Cyrillic spelling Квашук resulted in at least one Ukrainian family tree, the user melnyk_mariya99.

Translated with www.DeepL.com/Translator (free version)

There is Podolsk in Russia close to Moscow:
https://en.wikipedia.org/wiki/Podolsk
In Ukraina exists only Каменец-Подольский:
https://en.wikipedia.org/wiki/Kamianets-Podilskyi

Thank you Steve, does this mean I am unlikely to be able to trace him at all?

You should have a look at FamilySearch, they have a lot from St. Petersburg in their archives. It is best to always use the Cyrillic notation because of the indexing.

Квашук

I have tried but return no results at all with this spelling, sadly.

There are lots of Kvashuks in Ukrane: https://ridni.org/karta/%D0%BA%D0%B2%D0%B0%D1%88%D1%83%D0%BA

In Ukraine there is a historic teriitory calld Podillya, whre e.g. Kamyanets-Podilsky and Mohyliv-Podilsky are situated. Also Podolsk may refer to the Kyiv district of Podil (Podol in Russian).