Good question… as a non-linguist and non-Eastern European, my best try at an answer:
“Kapriz” in Russian is каприз, which translates roughly to “whim”. The suffix “ov” more or less means “of”, in the sense of being part of a particular family (same logic that causes so many English names to end in “son”). So, Kaprizov means something like “son of the whimsical guy”.
(So it’s enticingly close to meaning “son of the Wild one”)
The “off” in Kiprusoff also means “of _____” so the question is where the “Kiprus” comes from. Unfortunately there’s no easy answer. It is literally the Greek word for Cyprus, which is where Nick Kypreos got his surname. If that’s not the connection, it’s something obscure. It could theoretically be connected to каприз, but it would more likely be spelled Кипрус in Russian,
so there would have to be a bit of corruption involved.