Hey man!
Since you know very well what is your goal it's easier to find specific exercises to improve your score.
At first, this exercise is typical case which demands power. Power in sport is defined as ability of a muscle to quickly generate force over a short period of time. And because body work within physics laws, power = (force x distance) / time. Since distance is constant (your body - arms movement) you can manipulate force (high as possible) and time (short as possible).
Force is improved by strength training. In your case should be specific as much as possible. I would suggest you to doing push ups with hand positions like it is when you push the ball away from you. Start on knees, progress to normal pushup leg position -> elevate legs -> put weight plate on your back.
For decreasing time you have to speed up contraction of a muscle. This is done by adapting nervous system with plyometrics. Here I would suggest you two exercises. First would be plyometric push up. Since it have to be specific for your needs, hand position must be same as when you do push ups (for strength). Because of your specific position, full droping would be too big stress, start with wall push up. One step from wall, fall to the wall and in last time catch yourself and explosively push you away. Progress with being further to the wall, later get in position for normal push up and start again from knees. Drop yourself down, catch in last moment and explosively push up.
Because of very specific position be very well warmed up, streched and caution during working out this plyometric exercise!!
Second exercise would be next: get in position like you have to be on test, sit against wall and take light ball, which will have similar size than medicine ball on test; perfect would be basketball or valleyball ball, because bounce good. Make fast repetitive throws. Here your train just pure speed and reaction.
case for plyometric wall push up
That's it. Good luck on tests!
If anything unclear, ask ;)
BR, Gregor