score:2
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.