You can use word "shachta" for mines indeed, but I think this word sounds too modern for XIX century. It can be more authentic to use word "rudnik".
The word "priisk" was usually used for mines, where precious metals or gems were extracted. In some cases, they used word "priisk" for copper or lead mines.
"Farm" looks normal for cereal farms. Sure, they were not "farms" in the full sense of the word, as they didn't belong to farmers, but for the game it can be fine. You can also call them "uhod'ya" (owner's surname should be mentioned - "Vorontsov uhod'ya", "Potyomkin uhod'ya" and so on).
You can use word "kariyer" for mineral quarry or maybe "kamenolomnya" in case it extracts stone.
Word "rogati" is currently used for cattle farms. It will be better to use "rogatiy skot", which means "cattle" in Russian.
I really hope you can understand my English
