Light years is a unit of distance, not time, as in, the distance light travels in a year.

I'll just get to my actual question, if the universe is only so old, how far into the universe can we see since light still takes time to reach us?

Last time I checked, the observable universe is about 47 billion light years in radius. The difference is because spacetime is expanding at a rate faster than the speed of light. You'll probably ask yourself, doesn't that mean we wouldn't see anything, and the answer is no. It's a tad hard to explain over text, but if you picture the observable universe as a sphere growing at the speed of light, it's slowly capturing light that is farther and farther away. That's the entire idea of "red shift," since the space between us and the light source as moved away, the light is less energetic.