I have painted many cages for softbilled birds, like canaries and finches. I used black Rustoleum spray paint but first called the Rustoleum company and gave the customer service person the numbers off the bottom of the can. I asked about zinc in the paint. There was none. It dried very quickly but I still waited about 2 weeks before putting the birds in the cages. The paint must be used outdoors and allowed to cure so there are no fumes to injure the birds.
The paint held up very well but will not hold up well if you have hookbilled birds since they may chew on it, and will climb on it.
If the cage is rusty, you must clean all the rust off first, scrub it down and allow it to dry before painting. It is a process that will take several days if done properly.
Annie