A Nano would do the job, or an Uno, there's nothing special here unless you've not told us about future expansion plans.
But, power only the distance sensor with your Arduino. You will need to use motor drivers and relay boards, or solid state output drivers, for the solenoids and cylinders. Power should be from a separate power supply. Don't forget transient suppression diodes on the solenoids, they protect both your switching circuits, and the power supply itself.
Once you get a bit further along, reply here, in this thread, with a proposed schematic and parts list(include tech spec links for the solenoids, motor driver, and cylinders(what drives them, more solenoids?), and we can discuss what you've designed before you purchase anything.