Can I fight a charge against me if I was not given a breathalyzer test when I got arrested?

Miranda only has to be read if there is post-arrest interrogation. Most cops are trained to get everything they need out of your mouth "prior" to cuffing you, so Miranda rarely applies. With that said, you should hire a lawyer and get this relatively minor charge dismissed. By yourself you're going to get a criminal record and lose your license for a year, and it isn't worth it.

Unfortunately you are likely to lose on each of those points. You do not have the right to a lawyer before the tests (breath, blood or urine) are given. Even without a breathalyzer (and I'm not sure if you mean an 'intoxilyzer' or not), if you were driving in an impaired fashion, and the impairment in driving was due to ingestion of alcohol or a controlled substance, they can charge you with drunk driving, which is a different offense from driving with a prohibited alcohol content in your blood. Miranda rights are not very helpful. If they do not read them to you, and if you are in custody at the time they question you, then all that happens is that they cannot use your statements against you in court. But usually they don't need to do so Find a skilled OWI lawyer. It almost always helps.

You can, and should fight every charge. If they did not offer you a breathalzyer did they take your blood? If not it seems like they don't have much evidence. If they have your blood- well... they have your blood which is even a better way to determine your Blood Alcohol Content (BAC) to see if it is over .08%. If you answered questions to the police after they arrested you then your rights should have been read to you. For the most part, police ask questions before arresting you and so they don't need to read you your rights.

You can fight any charge, and you probably should. But know that breathalyzer tests are rarely given during a public intoxication investigation because, unlike DUI, the law does not state any particular blood alcohol content equals public intoxication. Also know that a failure to read Miranda rights does not mean a case gets thrown out.

Yes you can always fight the charges. The less evidence against you the better.