so, on the carb side since you have it opened up, the little black round guy is the internal carb fuel filter. pull it out, it'll need to be cleaned up or replaced, but be careful with carb cleaner, i think it can eat up the mesh screen.
on the plate side, there should be a check valve in each of the holes, which is how the pressure pulses move fuel through the fuel pump. if they get damaged or don't seal, the pump won't move fuel.
the carb you have is rather unique as it is a single carb that also has an accelerator pump (the little hoses running around the carb and the little black square pump with the linkage to the throttle arm). setting up the carb correctly for your 587 will be tricky as there wasn't ever a ski that came that way from the factory. the carb likely came off of a 96 or later gti or gts or gs with a 718cc engine. the accelerator pump works just like it does on old carb cars, lets the idle be set leaner and supplies a little squirt of fuel as the throttle cracks open while the fueling catches up.
the main and pilot jets and needle/seat and diaphragm is in the opposite end of the carb. you will want an impact screwdriver, plenty of carb cleaner, and some specialty tools to properly set up the low to mid range fueling for the carb. a pop off tester can be made from a bike pump, some pieces of pvc, some brass fittings and fuel line. you'll need this tester for various other checks and tests on the ski so its a worthwhile investment to make. until you have one, you can't properly set up the pop off setting with the needle and seat.
I'd recommend that you read the carb tuning stickies in this and the 2-stroke boat forum to learn as much as you can about these carbs. Also there are some good diagrams and explanations in the factory service manual, if you've downloaded it.