I am interested in applying to Sapper School. What do I need to accomplish to better my chances of being accepted?

Unless you have a chain of command that is really willing to put in a lot of work to make this happen, your chances of going are 0%. They currently only accept AOC/MOS: 12A, 12B, 12C, and 12N. As an 11B, you would need an MOS waiver request signed by an O-5 then routed to the engineer proponent for approval/disapproval. Most classes get filled with new 12As attending EBOLC from the RUMINT I've heard.

You are an airborne infantryman, not a combat engineer. If you want to do something special during this contract, you should start training for Ranger school. If you are set on Sapper, you should research what you need to do to change your MOS to 12B as soon as possible. Even as a 12B, there is no guarantee you will be offered Sapper.

I suggest you use your super internet sleuthing powers to actually find information about Sapper School qualification. I will give you a hint. Try Sapper Leader School and Fort Leonard Wood

There's a Sapper Handbook on the Sappee Facebook page you can use to study. I believe the recommended PT score is 270, but do your best to be higher. I'm training myself because I want to go to Sapper School too. Good luck to you. Look around YouTube videos and online as well to get as much information as you can. Some points of interest I've noticed are Land Nav (especially at night) and patrolling (especially in leadership positions), and knots.