Ok, couple of things to check. I'm not sure that trying to start it using jumper cables is a good idea, as you really don't get a good, solid connection with the clamps on jumper cables. But, since you said it has always done this, I'm going to say that the jumper cables are not the problem.
The clicking sounds like the starter solenoid trying to work, but I'm not sure. It's uncommon for them to allow any juice to flow, and they typically don't work at all when the solenoid fails. Some of these Japanese starters have the solenoid right on the starter itself rather than a separate starter solenoid like other tractors may have.
I would definitely pull the starter and have it rebuilt. A local armature or auto-electric shop should be able to rebuild it for you for $50 to $100, and it would be well worth the money. They can also tell you if the solenoid is bad (if it's on the starter), or if it is separate just take it in to them.
Those are my guesses based on what I saw. There is always the chance that the cylinders are rusted a little from it sitting outside, causing too much drag in the engine. It wouldn't surprise me if this were the case. However, again, since you said it was doing this before, I'm still leaning towards the solenoid and starter.
Another thing to check. These engines needed every ounce of juice that they could get from the battery. I would make certain that all of your battery wires, ground cables, etc., are new. Just replace them! I've seen MANY Cub Cadets of this era that the wiring was corroded so bad, inside of the insulation, that the copper wire just wouldn't transmit power. I've also seen them shorted out where they rub against the frame. With this engine needing every ounce of juice it can get to start, a little short, bare wire, or corrosion in the wiring can really kill the starting system! Just replace the wires!
Lastly, I'd make certain that your battery is good, and the biggest cranking amps that you can get in the size that will fit in the tractor! Batteries have come a long way since 1989 when that tractor was built!
As for your hydraulic valves, spray a little Kroil on the tops of the pistons and keep working the levers every now and then. They'll free up in a week or so. They just have a little rust from sitting.
Those are my recommendations, take them or leave them.
Bill
ito: Agree on the strong battery, good clean connections, etc. Sometimes when in doubt I boost the good battery with another one to be sure it's getting what it needs to start. A slow turning diesel engine will always have problems no matter if it's a lawn mower or a dozier. Just my 2 cents worth.