I’m a UK Citizen, and have a B1/2 Tourist Visa.

I applied for the visa as per the embassy guidelines as I was once arrested for Drunk and Disorderly. No conviction was ever made.

I attended the embassy interview and underwent a US Medical exam to prove I’m not an alcohol abuser and am in good health. This all came back positive and my visa was approved.

I have my visa now. But from what I’ve read, a Visa doesn’t guarantee entry. What are the chances of me now being denied entry by an agent at the border?

But from what I’ve read, a Visa doesn’t guarantee entry... what are the chances of me now being denied entry by an agent at the border?

Exceedingly small.

That disclaimer is there because the immigration officer (IO) may uncover grounds of inadmissibility that were not considered by the visa officer. For example, the IO may develop a suspicion that your trip is for criminal ends and that you managed to conceal this from the visa officer, or that you concealed some other criminal history from the visa officer, or so on. In cases like that, the US obviously wants the IO to be able to refuse entry.

It's also possible that the reason for your trip has changed, especially if it is a subsequent use of a multiple-entry visa. For example, if you show up with a B-1/B-2 visa and the IO discovers that you're planning to enroll in a university degree program, or suspects that you will stay for longer than six months without applying for an extension of status, you will not be admitted.

Like many legal disclaimers, this one is there to cover the possibility of refused entry even if it is very remote. In this case, it's very remote indeed. The IO will not normally second-guess a determination of admissibility made by a consular officer who has more time to examine more evidence about you than the IO has in the immigration hall. The thing that's most likely to trigger problems is the IO suspecting that there is new information from interviewing you that the visa officer did not have.

I would not be surprised if the IO asks you why you applied for a visa instead of using the VWP. If that happens, of course explain your arrest and the disposition of the case. They might put you in secondary inspection if they want to check your story against some file that they can't look up at the primary inspection booth, but I suspect they'll just stamp you in.

As the other answer points out, an arrest with no conviction does not make you inadmissible. As a comment points out, you probably could have used ESTA because of that. But you played it safe and got a visa. You appear to be going out of your way to be compliant. They should thank you for your honesty rather than keeping you out.

If you watch any of the documentary series about border forces in any country, you’ll see many cases of people being denied entry, even (or probably even often) when they held a visa.

Even if those documentaries are not representative of the whole spectrum of entry refusals, consider the cases they show. In most cases, it’s because they find out that there is a discrepancy between what the person said during their application and reality. The most frequent case is probably people coming on a tourist visa who actually intend to stay, live and work, usually disappearing under the radar as soon as they exit the airport.

So entry refusals with a visa are mostly to catch this kind of thing: you said one thing, and you intend to do so something else (or at least they suspect you do), or you weren’t quite honest about everything.

Here — at least according to your story in your question — you seem to have done everything you could to be above board (which is quite a change from many other questions here where people want to cheat, lie, misrepresent things to try to sneak in). Unless you misrepresented things, the chances of you being denied entry are quite small.

As you attended the check, and if your record is recently clean you should be allowed to enter.

The US GOV website provides a statement for this:

Can I be admitted into the United States if I have a misdemeanor or a criminal record (for example a DUI)?

The United States (U.S.) does not deny entry to persons that has a "Driving Under the Influence" (DUI) conviction. Although, if there are multiple convictions for this and/or other misdemeanors, you could be denied entry.

Generally, any convictions for drug possession can result in a denial of entry. If the conviction was long ago, you may have to contact the U.S. Embassy, Office of Consular Affairs in your country to obtain a waiver. Other misdemeanors may result in denial if they were recent.

A crime involving moral turpitude (CIMT) may be grounds to deny entry to the U.S. For more information, please visit the U.S. Citizens and [Immigration] Services (USCIS) website on General Categories of Crimes Involving Moral Turpitude (CIMTs).

You should be accepted for your honesty.

You're asking about a) the alcohol, and b) whether they will refuse you.

They won't refuse you for the alcohol. That matter was settled at visa application time. The only way it will rear its ugly head is if you present to the immigration officer evidencing that the settlement was a lie.

They will have done a basic screening for all the usual stuff (means, ties to community, etc.) That is why your admission is so likely. As always, they want to see that you won't overstay, seek employ (unless your visa permits), go on the dole, commit crimes, or be trafficked.

As far as some other reason, all bets are off. We can't guess. If there are 12,000 cigarettes in your bags, you're on your own!

When you try to use your visa, you may be sent for secondary processing because it is unusual for UK citizens to enter without using the visa waiver scheme. If this happens your passport will be taken from you and you will wait in a side room for the secondary check, this waiting time will obviously depend on how busy they are. A lawyer will probably ask if you need help, trying to drum up some business. Eventually when you reach the end of the queue, they will call the Embassy in the UK and establish that your visa is indeed valid, and then you are through. All the while you will be expecting extraordinary rendition to Guantanimo bay, but the worst case scenario will be that you are put on the next plane home. You will probably notice subtle little doors in the airport that lead to places normal passengers never see. Have a great adventure.