I only ever did the sample questions on the JLPT site, I've never been to an exam. So my answer assumes that those samples are the same level as the actual test.
With that out of the way, N5 listening is very basic. So you're behind on listening. By a lot. Safe to say that's your problem area. Which is good news, because
1. Listening/speaking is one of the easier things to learn about Japanese (especially when it's standard Japanese, like on these tests).
2. N3 listening is not that tough (again, based on those samples). It's much easier than understanding ANY native media.
So yeah, I think it's possible to get there, if you change the way you study...and spend more time getting good at listening, by doing various drills (lots of materials available, not gonna go through them), as well as (and this is important) immersion (comprehensible audio input, in this case...how you make it comprehensible is up to you: you can study the subtitles first and then listen to the materials without subs, or you can try and find stuff that's comprehensible without subs). Lots and lots of immersion. Hours and hours, every day.
Do that for a while, see how you progress. Then, when you get good enough at it that you're moderately comfortable with the N3 listening test, you can worry about catching up on the Kanji and grammar. Based on your results, you're already good at doing that, so it shouldn't be a problem advancing quickly, with full time study.
With that out of the way, N5 listening is very basic. So you're behind on listening. By a lot. Safe to say that's your problem area. Which is good news, because
1. Listening/speaking is one of the easier things to learn about Japanese (especially when it's standard Japanese, like on these tests).
2. N3 listening is not that tough (again, based on those samples). It's much easier than understanding ANY native media.
So yeah, I think it's possible to get there, if you change the way you study...and spend more time getting good at listening, by doing various drills (lots of materials available, not gonna go through them), as well as (and this is important) immersion (comprehensible audio input, in this case...how you make it comprehensible is up to you: you can study the subtitles first and then listen to the materials without subs, or you can try and find stuff that's comprehensible without subs). Lots and lots of immersion. Hours and hours, every day.
Do that for a while, see how you progress. Then, when you get good enough at it that you're moderately comfortable with the N3 listening test, you can worry about catching up on the Kanji and grammar. Based on your results, you're already good at doing that, so it shouldn't be a problem advancing quickly, with full time study.