Yesterday we arrived in Kazakstan. Not in Aqtau but in the New port of Kuryk, is 80 km south from aqtau (is still under construction). We didn't know about this connection.
We sailed with the Mercury 1 about 24 hours. ( we think that is shorter than to aqtau). The ship was mainly in use for shipping trains but they told us that they had some space left for 6 cars and a motorbike.
We went on board off the ship on friday evening and left the port of Alat Saturday morning around 9 am. We arrived at Kuryk at sunday morning at 9 am. But we left the port after 5 pm. Because it's a new port they aren't organized yet and they have no structure.
When we bought our ticket in Alat they said there was only a ferry on saturday or sunday, never on friday.
So our advice,
First buy the ticket from the first moment you arrive and stay camping at the ticket office because you never know when there is a boat. Because going to Baku and calling them doesn't work.
You really need to be persistent.