how is Zaroff able to finance his lifestyle?

Zaroff made wise investments that left him well off financially. He had no worries about money. He was rich.

"After the debacle in Russia I left the country, for it was imprudent for an officer of the Czar to stay there. Many noble Russians lost everything. I, luckily, had invested heavily in American securities, so I shall never have to open a tearoom in Monte Carlo or drive a taxi in Paris.