What is the verb to use when I want to say a festival becomes popular? For example: Christmas (verb) in the Middle Ages? (The time might be wrong, I'm just made it up)
The most idiomatic way to say this in formal or academic writing would be just the way you did in your question, using the verbs "become" for something that was not at all popular to start with or "grow" for something that was less popular to start with. So any of the following would work:
Christmas first became popular in the Middle Ages
Christmas grew more popular in the Middle Ages
Christmas became more popular during the Middle Ages
Christmas grew in popularity during the Middle Ages
You could also replace "become" or "grow" with a more colorful verb, as: "burst into popularity" or "burst onto the scene" to describe a sudden access in popularity.
There are informal ways of describing what you're asking for, like to blow up, but they are extremely informal and would sound forced in the context of a historical analysis. You are more likely to hear:
That DJ really blew up after his Albequerque gig.
The only 'one word' that comes to my mind that may fit your sentence is spread
To spread: To increase in range of occurrence; become known or prevalent over a wide area: The word spread fast.
Another simple way to write it would be to use a passive construction such as:
Christmas was popularized in the Middle Ages.
The active form of this verb requires an agent that does the popularizing. I realize passive voice typically isn't ideal, but if you really want to use the sense of the word popular, this might be the best way to do it.
Something else to consider: examples using "grew" and "spread" are using metaphoric expressions, so depending on your audience, you might want to be wary of these constructions.