If not, exit the game, sign out, sign back in, and then restart the game. If you see Automatron listed there, you’re good to go.
You can either buy it separately for $10, or get it as a part of the Season Pass. In spite of everything, Automatron has a lot to keep track of, so you might need some help and direction with where you want to go.
If you got Automatron, you probably want to get a handle over everything it has to offer. Automatron is one of the smaller bits of DLC for the game ( Far Harbor, which is the last bit of DLC that Fallout 4 will get under the first wave of the Season Pass, is actually the crown jewel of the bunch)- it adds robotic enemies to the game, and also lets you modify and create your own custom robotic companions. Bethesda are known for their excellent post launch support for their titles, so it should surprise no one that the first bit of DLC they prepared for Fallout 4, Automatron, is actually really damn good.