If I get vaccinated against influenza. Does my baby get any immunity from my vaccine, being breast-fed?

Passive immunity. Yes, you can pass antibodies to baby through colostrum in breast milk, but these will not confer long lasting immunity.
See below. The greatest degree of protection conferred by your vaccine is because you are less likely to get the influenza, and therefore not expose your baby. Have everyone around your baby get vaccinated too.
A little. Breast feeding transfers primarily IgA antibodies that help reduce germ adherence to/entry through to the tissues in the digestive tract. A small amount of IgG is passed, but not enough to be protective. The Flu shot helps you develop IgG (not IgA) to the flu & getting it while pregnant does protect newborns for a few months.Having mom immune reduces exposure risk.