If your looking for more areas to check out on that ski, here are a few important ones... Remove the grey plastic PTO cover at the rear of the engine and you should see a rubber boot with a grease fitting, make sure that boot is intact and not cracked and with a grease gun, pump grease in the boot just until you see the boot expand, then stop. the other grease fitting towards the rear is for the through-hull bearing and pump grease into it until you see it come out around the shaft.
Next, remove the pump nozel (4 bolts) and replace the jet pump oil (this is missed alot) on the nose cone, you will find 3 allen head bolts, crack them free and with a rubber mallet, gently tap the nose cone off. drain the pump housing oil (if its on a trailer, jack the front up so the oil can run out quicker) Inspect the o-ring on the cone, replace if necessary. Then smear a little lube on the o-rign before tapping the cone back on. Tighten the cone (dont kill it, just snug and an 1/8th of a turn) you'll see an allan head fill plug. remove and fill with gear oil (outboard lower unit oil works just fine-75W90) Inspect impellor and wear ring (you will need a feeler gauge)
I have a '91 GT same motor and it seems for some reason the bracket holding the top of the airbox to the head, the bolts always work themselves loose every season. be sure to tighten them up and use a little loctite on them. Inspect fuel system, drain and clean inline fuel/water separator located in the front of the seadoo (remove the front storage bin and should be located near the steering column. drain and inspect/replace filter.) Replace the inline oil filter and inspect lines. Be sure there are no air bubbles in the line, or you will have to bleed the oil pump.
Lube the steering cable and throttle cable. (for throttle cable i usually hold it WOT and put a couple drops of ATF (tranny fluid) down the cable. works amazing. clean/inspect plugs, check compression.
The 587 is a bullet proof setup. My 91 GT has had the ball bag run off it, tubing, skiing, towing boats, wild rides, subing it (with reverse) family bought in '94, was a rental ski from 91-94. Original motor, original carb setup, original impellor/wear ring (will need replacing soon), new starter/solonoid, new plugs every year, did away with fuel water separator, using an aftermarket unit (brass top with glass bowl) no special maintenance.... just did a compression check yesterday, 150 psi both cylinders. just goes to show how a little routine maintenance goes a long way...
