"They will get married next year in Portugal."
Translation:Ils se marieront l'année prochaine au Portugal.
To express going to a country or continent, the following rules apply:
Singular masculine countries/continents that do not start with a vowel sound use "au"
Singular masculine countries/continents that start with a vowel sound use "en"
Singular feminine countries/continents use "en"
Plural countries use "aux"
In this case, Portugal falls under the first category, so "au" is correct and "en" is not. For cities, usually you use "à" on its own, not "au", though there are a few exceptions where "à la" is used. It gets even messier with states and provinces; usually "dans" is used, but for ones that were historically independent like Quebec or Texas, you follow the rules for countries above.
You can see here for more details and examples, particularly starting at the "Countries and Continents" section.
You can see here for more details, but the simplified answer is that if you just want to say a specific year (e.g. "the year 2020") or treat "years" as just a unit of time (e.g. "12 years old"), you use "an". In most situations where you're describing a year or considering the duration of the year, you use "année". In this exercise, "année" is used since the marriage event is happening sometime within the duration of the next year.
The exception is the demonstrative adjective "cet", in which case "an" is used
No you don't say cet an but always cette année. I'm sorry to say that the information given with your link is wrong. Cet an sounds a bit odd.
If you're describing the year with an adjective like "next", you use "année"
Not really, l'année dernière and l'année prochaine are also equally correct.
You're right, I agree with the first point. My statement was an oversimplification. As the link states, when you are using the demonstrative to say "this year", it's always "cette année". I think it's still possible to say "cet an que..." as the link states, but I do also agree that it's a weird and at best uncommon construction.
I think you meant "l'année prochaine" and "l'an prochain" in your second point, but again you're right that there are contexts in which both are acceptable.
Since the original link and comment I posted was potentially misleading, I will edit it to be more accurate shortly - thanks for your feedback.
that masculine/feminine correspondence also applies more generally to contrast points in time and periods of time:
le matin / la matinée le soir / la soirée
(fwiw, there's an equivalent "une après-midi", but that's very much a literary thing: the only time i can remember encountering it in Zola.)