Greetings.
2 teams work in Voronja cave every year: Ukrainian Speleo Association (Ukraine) and Cavex team (Russia). Dates: july-august.
Ukrainian cavers work mainly in Non-kujbyshevskaja series (-1700).
See cave profileSome pics from 2011 expedition:
http://speleoukraine.org/Forum/Read/thread_id/6Expenses.
It's usually cheaper to go with ukrainians. This year expedition consisted of 4 men. Expenses: $600-800 per person.
Add to that plane ticket to Moscow (Russia) or Kyiv (Ukraine) and plane/train ticket to Sochi (south of Russia, abkhazian border).
Find out whether you will need visa for Abkhazia or Ukraine. You would definitely need visa to Russia.
Looks to me like you would total around $2 000.
You can reach Yurij Kas'yan: kasyan_yura at mail.ru for details.
Regards,
Salavat