If you can afford a private attorney, you do not qualify for a public defender, but courts do not like to let people go unrepresented at court in a criminal proceeding. An arraignment is a simple procedure simply involving entering a plea of "not guilty" and setting the dates for your settlement conference and preliminary hearing. You *could* do that on your own, but it's better if you are represented and the court will prefer it.
You're in luck because Riverside county courts are handing out extensions like candy right now. When your name is called, you need to advise the judge that you are in the process of obtaining private counsel. You need to be able to give him a date in the not-to-distant future where you will have the counsel. Ask for a continuance pursuant to PC 1050. Since you have the right to a speedy trial, you will need to waive your rights to the time limitations for the purpose of the continuance. The judge will give you a new date to return for your continuance, and will probably set a tentative date for the FSC and preliminary hearing. Be SURE that you have counsel by the new date.