Can I work out a payment schedule for my patent? How?

Depends on the credit policy of the law firm you are working with.

Maybe - it is up to the lawyer or patent agent you use to do the work. Some law schools have IP Clinics that provide assistance to inventors with limited funds. Most US utility patents can be prepared, filed, prosecuted to allowance for about $10,000 or more.

To the best of my knowledge, the USPTO requires that the entire application fee be submitted at the time of submission and any other mandatory fees during the examination must be paid at the time they are requested. You may be able to work out a payment plan with the attorney, agent, or other expert who helps you prepare the application. If a "We'll help you get a patent" company offers a loan, read the fine print very carefully before committing yourself. All these expenses have to be paid, whether you get the patent or are rejected.