It's somewhere between the two -- and depends on what school you attend.
Linear algebra is used a lot, particularly in statistics/econometrics. You need to be comfortable with matrix notation, functions on matrices, inversion, etc. If you have to think about the notation or matrix operations, it will be really hard to keep up with the actual content.
Real analysis is a part of much of the first year sequence, especially micro. You will prove that there exists a utility function that represents preferences. You will prove that function is continuous, LNS, etc. You will use direct proofs and proofs by contradiction. You will use concepts like "open balls," and constructs like, given delta, there exists and epsilon such that... There's a lot of notation (symbols for things like "for each" and "there exists" that isn't difficult, but will make it hard to follow the professor on the board if you don't know the symbols being used.)
Now, some of this material will be covered in "math camp" or a math class during your first year. But typically that class is just a quick overview or refresher -- if it was your first time seeing the material, you'd probably need to do quite a bit of work outside of class to actually learn it on your own. And that will be time you just don't have first year. Real analysis and linear algebra are both tools that are supposed to make the intuition clearer. If you don't understand the math, then instead of focusing on the economics, you'll be stuck paying too much attention to the math.
(Note: this all pertains to top 20 or top 30 schools. I don't know as much about the first year classes at other schools. Also, I'm not advocating taking tons of extra math above and beyond those classes, but I do think they are useful in the first year as well as important for admissions. And it's a great question, because the focus should be on how to do well, not just how to get in.)