Are there nonstop flights from Moscow (VKO) to Batumi (BUS)?

Yes. Azimuth, Georgian Airways fly this route. Flight time is about 2h 22m.

What is the fastest connection from Moscow (VKO) to Batumi (BUS)?

via Sochi ≈ 2h 53m, via Krasnodar ≈ 2h 53m, via Mineralnyye Vody ≈ 2h 56m (estimated flying time, excluding layovers).

from 15,000 ANA miles (economy, one way, based on the published award charts).

Vnukovo International Airport (VKO) to Alexander Kartveli Batumi International Airport (BUS)・1,586 km

Nonstop flights available (2 airlines). Flight time about 2h 22m. Nonstop return flights also operate.

With milesANA 15,000 miles+ (via IST)Economy, one way, estimate. Tap a connection for miles by cabin.

16 one-stop options, sorted by estimated flying time. Tap one to see the airlines on each leg and booking links.

Times are estimated from distance. Check schedules, frequency and availability with the airline or a booking site.