Why do Aya and Youko consider going to a ramen shop alone so amazing?
Shall I assume that these characters are in high school or younger? In that case, yes, it not so common for high schoolers to hang out at a ramen restaurant compared to a fast food place like McDonald's or a family restaurant like Jonythan's where they can stay for a long time using the drink bar (all-you-can-drink soft drinks/juice/coffee/tea), study there, chat with friends, etc. Ramen restaurants are generally not upscale and, as you can see in the illustration, strangers may sit next to each other at bar-like seating, so this makes it convenient for middle-aged salarymen to eat lunch there alone (again, this is shown in the illustration).  Ramen shops are not very conducive for having conversations, hanging out, and staying past when you empty your bowl. Thus, it's not a place that usually has a family or young people vibe.

If these characters are university students, it is less likely that the Japanese girls would consider Karen fearless, because university students tend to want to eat cheap and quick food, so ramen joints work for that, but it is still that case that many female college students wouldn't go to a ramen shop alone, and possibly wouldn't even go there in a group unless that group includes a male friend, because they will assume that the place will be full of college guys and older men. It's more common for Japanese university girls to eat at regular restaurants and cute cafés.