Vilna is an old Russian name for Vilnius, the capital of Lithuania. It was also the name of Vilna Governate, a Russian territory that from 1795 to 1915 covered the south-east of modern Lithuania (and beyond into Belarus). So maybe we can start by assuming that Eiszuk was in the part of Russia that now Lithuania (this is just an assumption, it would be nice to have other evidence).
As you probably know, Lithuania has been controlled by various countries over the years. Many languages have been spoken in the area. Places have been renamed, and are spelt and pronounced differently in each language.
Wikipedia has a useful (but maybe incomplete) list of Lithuanian place names in various languages.
There's no direct match for Eiszuk, but Eshishuk is shown as the Yiddish version of Eišiškės. The pronunciation of Eshishuk could be be fairly similar to Eiszuk (and a Yiddish speaker would not be able to spell it in English, or Russian).
Wikipedia has a detailed page on the location and history of Eišiškės. It's right on the modern border with Belarus, so would have been near the middle of Vilna Governate. The page mentions that an alternative Yiddish spelling is Eishyshok. There's a much more detailed page, with lots of photos, at Zydai.lt (in Lithuanian).
You'd need other evidence to be sure, but I think it's quite likely that Eiszuk is this place that has been known, at different times and in different languages, as Eišiškės, Ejszyszki, Эйши́шки, Ejšiški, Эйшы́шкі, Ejšyški, אישישוק, Eshishuk, Eishyshok and Eiksiskendorf.