Can I sue my employer for firing me before I was convicted of a DUI?

Unlikely you could sue unless you are part of a union. Employees without a union can be fired without any cause or if there is cause you cannot be fired for discriminatory reasons (race sex, religion). You will probably lose your job when the automatic suspension which lasts at least 30 days begins.

Probably not, unless you have a very specific employment contract or are represented by a union. Most employment is "at-will" meaning that under most circumstances, you can quit or your employer can fire you without any repercussions. If you wish, you may want to contact an experienced labor or employment law attorney for a face-to-face consultation and give him/her all of the facts surrounding your situation. He/she would then be in a better position to analyze your case and advise you of your options.

In general, unless an employee is civil service, in a union, or has a written employment contract, they are an 'at will' employee that can be disciplined or terminated any time for any reason, with or without cause explanation or notice.