My suggestion is that you NOT assume that the judge would grant you a continuance if the court date is tomorrow. There is no set rule for these things, but in general judges are not inclined to grant a continuance, particularly if a witness has been subpoenaed for the court date.
My suggestion is that you hire a DUI lawyer TODAY and go to court with him/her tomorrow morning. At least you would have some opportunity to prepare for the trial and the privately retained lawyer can then ask for a continuance on your behalf. If the continuance is not granted, you can then have the privately retained lawyer represent you.
Some court appointed lawyers do a good job for their clients. (In courts where there are full time public defenders, they usually are very dedicated and experienced.) However, court appointed lawyers are paid $150 per case -- not per hour -- and so it is often the case that they do a very minimal job representing criminal defendants.