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.

It's been well documented to have a droopy lid after cataract surgery; best to wait up to one year to make sure it stays stable. Sometimes it can improve with time.  You should get an evaluation by an oculoplastic surgeon to see what your options are. Best of luck, 

Droopy eyelids after eyelid surgery may be due to ptosis of the levator muscles of the upper eyelids. The ptosis may have been present before and know is unmasked. An evaluation with your eye doctor is important to rule this out. 

I'm not sure why you believe cataract surgery caused droopiness of your eyelids but if that is the case, you should be evaluated by your ophthalmologist and an occuloplastic surgeon to properly diagnose the problem since you didn't post any pictures.

That
is a great question.  In fact, it is very common in my experience for
someone who undergoes cataract surgery, to be diagnosed as having ptosis, or a
droopy eyelid that is also impacting their vision. We all expect that at some
point we may get cataracts, but fewer people realize that droopy upper eyelids
are a true vision concern and one that we can often address often through
insurance authorizations.  As a surgeon, it is very common for me to have
patients who undergo both surgeries in their quest for the very best quality of
vision and quality of life.

Ptosis after cataract surgery is corrected with levator surgery to reattach the levator muscle to the tarsus. The ptosis sometimes resolves, so it is wise to wait 6 months before committing to surgery. Best wishes with your decision.