What are the possibilities to get from Yerevan to Batumi?
Is there any direct option?
How long does it takes and how much does it cost?
Besides taking the train as @Karlson points out, there are also multiple daily connections by minivan between Yerevan and Tbilisi. I think this took about 7 hours last time I did the journey.
Then, from Tbilisi there are trains and buses that can take you to Batumi. I seem to remember there are overnight trains, but a bus is much faster.
I don't remember the cost.
I also don't think there are direct connections between Yerevan and Batumi.