There are so many ways to look at this, and I had a good think upon it. That's why language is fun.
I agree that 'popular' generally means well-regarded whereas 'prominent' means most conspicuous, or outstanding. That's the take a way. Here are some scenarios. Neither are definitive but just exemplative [Safari is convinced this is not a word, I disagree].
Consider a high-school class of 100 people.
There's Amy. Plain Amy. Everyone loves Amy, she's really nice and she'll go out of her way to help you. She's really popular, i.e. of the people. But Amy is just a normal, nice girl, you wouldn't pick her out of the crowd.
Also in the class is Duke Fritzenberg of Moravia. No one even really knows who he is, except that he's a big deal. He comes too school in a very expensive car, dressed in fine linens, and obviously has the power to order rare peacocks for lunch. He is prominent. He stands out. He can do things.
Of course, Amy can do things too. . . Just, not peacocks.
Imagine the face, any face, of an actor, for example. This actor has the most beautiful blue eyes, piercingly gorgeous, you can't look away. Everyone adores him for his eyes, they are legendary, and he is very popular. That's evidenced by the fact that he is very high paid actor, perhaps, in part, because his eyes are so popular.
He also has a gigantic mole on his left cheek. The kind of mole that you notice only a split second after his eyes. Without the mole, he would be equally as popular, but when you see him, that mole is nearly on par with his eyes. That mole would be prominent, but also in the context of a popular face.
(The issue at hand)
In terms of sports, popularity and prominence are essentially the same by way of the free market. Soccer / Association Football is the most popular sport in the world. By that definition, is is preeminent, or prominent, i.e. the sport with the most coverage across the globe. Put another way, if an alien species were to land on Earth, they would notice a lot of people kicking balls into rectangular boxes. That would be the takeaway.
In the United States, because of the way various sports are structured (in terms of number of games, number of players, television time, salaries), it's harder to decipher. Baseball is, emotionally, part of the American identity. In that way, it's very prominent. Millions of kids grow up swinging a bat or throwing a ball, so it's also very popular. But lots of people all around the world play baseball, and cricket, which is vaguely similar, is likely the second most popular sport. American Football is unique, VERY prominent, and VERY popular. Golf is extremely popular in that many, many people play it (like angling, as per James K), but not popularly watched (also like angling).
In general, you wanted to find the right word. I think, for your usage, it's 'popular' -- you might use 'prominent' if you wanted to suggest that the sport had a particular force, a particular power, in a culture, perhaps incongruent with its popularity.
That is to say, everyone likes the popular kid, but the prominent kid, though popularly (well) known, controls the situation and bosses everyone around.
 In an odd turn, the 'popular' girl in school is usually secretly, or not so secretly, despised. So it's a pretty bad analogy. . . 'Popular' is a tough word. Donald Trump might be called a populist, and though he's is also prominent, he's also not very popular.