Planning on visiting the Azores next summer and I figure a little Portuguese linguistic training would not be amiss. Pretty much all the Portuguese language aids I can find focus on the Brazilian flavor. As I understand it, The Azores are a bit more…rustic…than their mainland neighbors in Portugal and their accent is notably different. With that being the case, how goofy am I going to feel when I hit the ground with my broken Brazilian? No problem? Or would I be better off throwing a bizarre accent on my English and just talking louder?
No problem at all. My Portuguese training prior to going to Lisbon with the State Department was six months of intensive Brazilian (there was no continental class on the schedule at that time). They understood me just fine. The only problem I had was understanding the Portuguese accent, as many pronunciations are different, i.e., the word for “day” is “dia”. In Brazil, it’s pronounced “gee-ah”; in Lisbon, it’s pronounced “dee-ah”. Doesn’t seem like a huge deal, but when someone is speaking at a normal rate, all those little differences create a sort of congnitive dissonance. But I knew Brazilians who had no problem understanding Lisboans, and vice versa.
I think Inigo’s question focuses more on the question of whether the Brazilian dialect will prove problematic in the Azores, not on the mainland.
Anecdotally, I can tell you that my sister-in-law, who is from the Azores, has Brazilian friends with whom she is able to speak in Portuguese. They found it a bit slow going for a few minutes, but were able to get comfortable pretty quickly. There are occasional bursts of amusement at some misunderstandings due to dialect differences, but nothing serious.
Most of my SiL’s family have said that they find Brazilian Portuguese to be spoken very fast, but I suspect that unless you achieve fluency before you go, that’s not likely to be a problem.
I can support Cerowyn’s anecdote with another one - I used to work with someone from the Azores. He could converse with Brazilians pretty much okay, but was also able to point out many many differences. As would be expected, he said it was much different from “Iberian” Portuguese, which he had no trouble with, but it was still intelligible.
In short, you should be fine.
There is definitely a strong accent difference between Continental Portuguese and Brazilian Portuguese. Indeed, there are some pretty good accent differences between regions in Brazil.
Just like American/UK English, there are often words that have different meanings; for simple tourist interactions, this should be OK, but be forewarned.
Example: I was told that the Brazilian word for Papaya, mamão actually means something rude in Portugal. Similar to how American “Fanny” is hilarious to Brits because it refers to a woman’s girly bits.
Another example: my wife was puzzled when she took her NJ driver’s exam in Continental Portuguese twenty years ago: they were talking about travão, which she never heard of, and it turned out to mean “brake”, which Brazilians call freio.
On a deeper level, there are some fundamental forms of language that differ. (Disclaimer: I have spoken Brazilian Portuguese for decades with Brazilians, but have rarely spoken with anyone from Portugal, if ever.)
The “ing” form of a verb is used differently:
“I am speaking” Brazil: Estou Falando Portugal: Estou a falar
I believe the usage of tu vs. você (forms of “you”) differ as well – the former is rarely used in Rio except in incorrectly formed colloquial speech (often mixing tu with the wrong verb form). I think tu is actually used properly in Portugal.
I also think Portugal has a greater usage of direct/indirect objects buried in verbs, kind of like in Spanish, where they do all of that mo-lo-lhe-la-blah stuff. Those are somewhat uncommon in Brazilian speech.
Add to all this the fact that the Brazilians are an effusive bunch, and very demonstrative, whereas the Portuguese can be fairly dour. When Brazilians speak, the words actually leave their mouths, while the Portuguese sort of chew on them for awhile and let one escape from time to time. It was very frustrating for me.
My wife and I spent our honeymoon in the Azores here.
What island are you going to? We spent our honeymoon on Sao Miguel. The above mentioned Santa Rosa is available for rent if you’re interested. 
I’ve also been to Rio and was told the language difference was akin to New York English compared to Georgia English. Words are the same, pronunciation and cadence are different.
You know how Americans love British accents?
It’s the opposite direction with Portuguese – folks love to hear the carioca accent of Rio de Janeiro 
. For that reason alone I’d say go and learn Brazilian Portuguese. The Rio accent is music to the ears!
So is their music, for that matter. I love me some Amalia Rodriguez, but a bit of fado goes a long way, whereas samba is a delight.