If my baby was really fussy after the last year’s flu shot, is it okay for him to have one this year?

Yes. General fussiness certainly is not fun, but does not fall under the criteria of a dangerous reaction. And it is still "better" than a serious, potentially life threatening case of the flu.
Yes. Any child can be fussy after any vaccine. This is usually mild, lasts less than 24 hours, and can be relieved by giving acetaminophen. The flu shot offers excellent protection against flu, which is a serious respiratory illness that leads to pneumonia. Tell your doctor about previous reactions when your child is vaccinate, for advice on how to manage discomfort.
Yes. Fussiness is not a contraindication to a flu shot. It is not uncommon for babies to react with slight irritability, fever, soreness or body aches after the influenza vaccine. The reaction generally improves within 1-2 days and a dose of Acetaminophen may help to alleviate the symptoms if needed.