How can droopy eyelid be corrected after cataract surgery?

Generally it is best to let the eyelid heal 6 to 12 months because the ptosis can recovery in many individuals. However, once this time has past, if the ptosis is not resolved, then yes surgery is what is used to fix the ptosis.  This work should be done by a fellowship trained oculoplastic surgeon.  You ophthalmologist should be able to refer you to the appropriate surgeon.

As someone who does both cataract and lid surgery, I almost always recommend doing the cataract first if bothered by both. Many patients have both.
Wait a few months after cataract surgery then proceed with any lid surgery you desire.
The appearance of the lids is often more obvious if you don't have glasses on to hide them.
Droopy lid can mean a droopy brow, low lying lash line ( the sleepy look) or ptosis, or just extra-lid skin which is fixed with a blepharoplasty. You may have a combination of all those things.

Eyelids that become persistently droopy after cataract surgery (or with aging of the eyelids), can be corrected surgically to repair the levator mechanism in the eyelid.  This mechanism normally lifts the eyelid, but can become stretched out with cataract surgery, and with aging in certain individuals.  A consultation with a Board Certified Plastic Surgeon is recommended to review your options.