I feel like for such a simple question, there are a lot of angles you can look at this from. I am going to talk broadly about the development of DUI laws in the 19th/20th century; I do not have the knowledge to approach this with a view to talking about the enforcement of criminal law in the American West, but would be very interested to hear a response from someone who does. If that was more of the angle you were going for, OP, then I apologise in advance, but if you want to look at this question from a literal, legal perspective then let's go.
I believe the earlier comments in this thread have been deleted, but I'm going to briefly get one thing out of the way that another poster mentioned: when we're talking about a "DUI" offence involving a horse or any other animal-drawn vehicle, it is irrelevant whether or not the animal is sober. If the human operator is intoxicated, and the law encompasses horseback riding, it's no defence to say that the horse was only drinking water the whole evening and he can keep on walking in a straight line even if you're sideways in the saddle.
The answer to any hypothetical legal question is always going to hinge on the time and the jurisdiction. For the purposes of this specific question, it's also going to become relevant whether our cowboy is a messy or habitual drunk or whether he can keep things discreet. Time-wise I'm going to assume we're talking about the 'Wild West' era, so let's say roughly from 1850 to 1920. Having a relatively late ending date means that at least some US states will have enacted specific DUI laws by then, even if it is starting to move us away from the classical image of the frontier cowboy. Generally speaking, DUI laws predate the invention of the automobile. By the second half of the 19th century, there were statutes on the book in some countries that explicitly made it illegal to operate a horse-drawn vehicle, ride on horseback, or drive herds of livestock on public roads while intoxicated. UK law is much more in my wheelhouse than US law I'm afraid, so I hope you won't mind if I cite a UK statute by way of example: the Licensing Act 1872 s 12 creates an offence whereby -
^(Every person who is drunk while in charge on any highway or other public place of any carriage, horse, cattle, or steam engine...shall be liable to a penalty not exceeding forty shillings, or in the discretion of the court to imprisonment for any term not exceeding one month.)
A couple of things to note here: first, the reference to 'highway' or 'public place' means that you were okay to get trashed and ride your horse or buggy around on private property (at least from a criminal perspective, you're still going to be civilly liable if you damage someone's property). Secondly, this offence is found in a liquor licensing statute rather than a set of 'road rules', or enacted as a stand-alone law as we might expect today. The same statute also contains an offence of being drunk in a public place, roughly equivalent to our contemporary public intoxication/drunk and disorderly laws.
The legal system in the UK is fundamentally different to that of the US - because the latter is a federal system of government, DUI laws were enacted on a state-by-state basis, whereas the English statute cited above was (and remains) in force nationally. It also has the benefit of using a term like 'carriage', which can be interpreted liberally and creatively to refer to other wheeled vehicles as they come along (like a mobility scooter!). DUI laws in the United States, on the other hand, tended to be enacted in specific response to the proliferation of cars on the road (from 1906 onwards), and consequently are more likely to refer specifically to an 'automobile' or a 'vehicle' or 'motor vehicle' - if this kind of language is used it will not encompass our lone cowboy on his horse.
That doesn't mean he's necessarily home free, though. I mentioned earlier that the Licensing Act also creates an offence for simply being intoxicated in a public place - if our cowboy is being disruptive or behaves in such a way as to draw negative attention to himself, then it's entirely possible that he could fall afoul of this law regardless of whether he is on horseback or not. Laws criminalising either public intoxication in and of itself, or public intoxication in conjunction with some kind of antisocial behaviour (some of which may be roughly analogous to some contemporary disorderly conduct/public nuisance offences today) were absolutely in force across the United States in the second half of the 19th century, although the exact elements of the offence and whether it was found in statute or common law again differed on a state-by-state basis.
So if our cowboy is riding on private property, or riding along a public road in a state where DUI laws were specifically drafted with automobiles in mind, and he is not creating any kind of nuisance that would justify him being apprehended for drunk and disorderly conduct irrespective of whether or not he is on horseback, then ride on, buckaroo.