The charge would probably be 'public intoxication'.
Not in most states. You need to be "Operating a Motor Vehicle" and a horse is not a motor vehicle.
If you do some research on the web you can be armed with as much knowledge and info as an expensive traffic attorney. You will be able to argue your case and maybe get a reduction or dismissal of the ticket.
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Another View: You may not be able to be charged with DUI, but you may be leaving yourself open to a charge of Public Intoxication.