At my local academy the Physical Fitness test consists of Push Up's, Sit Up's, and a One Mile Run. I would say focus on those when you are actually close to joining (or if you're even accepted for that matter).
Since you don't like Sit Up's I'm going to assume you're either over weight, or built like a twig (I get it I've been there, that's why I feel qualified to answer your question) focus on losing weight/gaining strength first before targeting any specific exercise.
I recommend getting on a program, and keeping a log, regardless of your goal (weight loss/strength gain/hybrid) you're going to have to follow some kind of diet. This may seem like a lot of work at first, but trust me over time it just becomes part of life, and you get the log down in under two minutes.
Again due to your question regarding basic body weight exercises, I'm going to assume you don't know the first thing about working out, so I'm going to set up a very basic program for you, that I recommended you follow for at least a month in order to kick start your strength gains, and so you get in touch with your bodies bio mechanics.
Here is the program, I recommend you walk (briskly) daily for at least one hour, and endurance gains. (again this is specifically for starting strength). If you don't recognize any of the exercises, google is your friend.
-Monday, Wednesday, Friday
. Walk
. 3 sets of Pushup position: This is only holding the position, do this for a minimum of one minute per set.
. 3 sets of 20 Body weight squats
. 3 sets of 10 Pull Up Negatives: This is only the decline motion of the pull up,do it as SLOWLY as possible.
. 3 sets of 10 Chair Dips: Use a chair to perform Dips
. Rest 20-30 seconds between each set.
-Tuesday, Thursday, Saturday
. 1 Hour walk
. Catalog everything, what you struggled on, and what your improving on
As for the diet, it's very simple: if it comes from the ground, or was alive at one point, you may eat it (Beef,Pork, Fish, Poultry, Vegetables, Fruit, and Legumes), eat as much as you'd like (until your satisfied) and as many times as you'd like, you should never be hungry, and you should never be stuffed. Also try to stick to drinking only water and 100% Fruit/Vegetable juice. It doesn't need to be anymore complicated than this (if your goal is not weight loss add complex carbs like potatoes,yams, wholewheat bread, oats ect.)
If you decide to take my advice, and see positive results (or the workout becomes to easy before the month is up) feel free to PM me, and I will help you out.