|Congratulations on joining the JH family. Despite its reputation, a JH is actually about as reliable as any other car of its era, provided that the owner performed at least minimal periodic maintenance. Once a car is left to sit, though, maintenance is generally neglected thereafter, and problems will arise.
Hopefully by now you've gotten your alternator problem straightened out. For the oily #1 spark plug, such problems often go away once the car's been driven regularly for a few months. If not, it's a lot less expensive to change the plugs fairly often, than to have the head serviced, so if the car remains driveable you may wish to tolerate the problem.
If you get a copy of the shop manual, and check everything on the list of scheduled maintenance items, you should find just about anything that might currently be wrong with the car. Keeping up with the maintenance schedule, even on a casual and intermittent basis, will generally prevent future problems, particularly if you follow up on the 'hmm, that's odd, why does this part look different today?' sort of stuff.
Examine the timing belt regularly. Replace the belt in accordance with the maintenance schedule, or if it exhibits damaged teeth, cuts, or severe abrasions. Be suspicious. If the belt should ever break, the engine WILL bend valves, necessitating expensive and extensive repairs.
Change your brake fluid every couple of years, to avoid having your brake system rust internally. Change the fluid more often in humid and coastal areas.
Use a high-quality 10W40 oil if the average temperature is below freezing, and 20W50 oil when it's warmer. Every time, without fail, check the oil gauge and ensure that you have oil pressure before driving the car. Keep the oil level between the 'full' and 'add' lines of the dipstick at all times. You may find that the engine generates a fine cloud of oil mist if the oil level approaches the upper line. If this occurs, withdraw some oil, or let the engine get rid of it.
The thermostat is a bit special. If the wrong type is installed, the engine WILL overheat. If the engine overheats to the point of boiling away much of its coolant, expensive problems are sure to follow.
Change the coolant every two years. This prevents internal corrosion, and helps keep the expensive water pump in good condition. If you see rust in the coolant, expect problems. Any antifreeze intended for use in aluminum engines will suffice, but avoid the orange stuff if you can. Coolant is toxic to pets and other small animals (including children), so dispose of it properly.
Check the coolant hoses every few months. Good ones should last 4 years, and usually longer. When an extensive network of tiny cracks is present near the ends of a hose, where it flexes most, you can expect eventual hose failure. It's prudent to replace all the hoses at the same time, in conjunction with a coolant change.
All the cross-head screws in the car are Pozidrive screws, not Phillips. A Phillips screwdriver will eventually ruin the screw heads. Pozidrive screwdrivers and replaceable bits, depending on your preference, are available from several online vendors, such as McMaster-Carr.
Have a can of aerosol penetrating oil (such as Liquid Wrench) handy whenever you approach the car, and apply a squirt of the stuff to any hardware that happens to catch your attention. If you do this regularly for a few years, you should never have any problems with frozen bolts.
Good luck with the car, and have fun.