For your electric starter, before rebuilding it, check your battery cables. If your starter is 6v, you need 00 minimum cables to ensure sufficient current reaches the starter. The battery, cables, solenoid, and clamps should be in good condition, with all connections bright, clean and tight. For 12v, I'd go with 0 cables personally - same other conditions apply. Some folks will use 12v on a 6v starter - your risk there is that it can be harder on the Bendix assembly but many folks report no issues. I"m personally a fan of keeping it stock but do what you need to do to get it running how you like.
You need all of that regardless of whether the starter needs rebuilding. You can make the cables yourself to get the length right - shortest run with adequate routing so the cable is not a danger to itself or peoples' feet. If that doesn't perk up the starter, then go ahead and send it out for rebuilding. If you would like to check things in the starter motor yourself first, check that the brushes are not worn out and move adequately in the brush holder. Also confirm that the bearings are in good shape. If the bearings are dry/on the way out, that can affect its ability to start the pony. If all that looks ok, then send it to the shop.

For the pony itself, check the compression real quick. If you haven't recently changed the oil, do so. Can sometimes "cure" a poorly starting pony with a change of oil. Assuming it's in good shape, then two things will get it easy to start - carb in good shape, and magneto/ignition in good shape. Given that 90% of carb problems are ignition-related : ), start with the mag, plug wires, and plugs. Replace the mag condensor (usually starts showing its weakness when the pony is already hot), and confirm that your plug wires are solid copper core. If not, go ahead and replace them immediately. You can buy new plug wires on the web, or make them yourself with some solid core from the auto shop. Might as well put new plugs in since you're there. You can get an OTC plug checker that you can use to confirm that the spark is bright and hot. The checker is purposely set to make it harder for the spark to jump, exposing any weakness in your ignition. Also verify that the grounding switch does actually open reliably. You can easily isolate it by removing the grounding wire from the mag to rule it out. Check and clean the distributor rotor from the mage and the inside of the cap. Clean with a dry clean cloth.
If none of that helps and you've verified the timing, then go to the carb. If the pony starts easily but dies, verify that you have sustained full flow from the tank. Disconnect the fuel supply tube from the carb and let it run full flow from the tank for at least 30 seconds. If the flow drops off, you need to identify where the blockage is. Note that foreign objects in the tank can obstruct the flow sporadically, but they may be very difficult to see in the tank. Thorough method is to remove the tank, drain it, and give it a good shake and inspection to confirm it's clean. If you do have good sustained flow, check the fuel bowl float level. Too low and it can be hard to start. Too high and it risks flooding (but you would have seen that by now). If you get a carb kit, you can disassemble the carb and carefully clean all the passages and jets/tubes. I'm not familiar with the carb on the 2U but the 5U carb has a couple of passages that are vulnerable to blockage, but once cleaned, works very well. The two tests I like are how many pulls it takes to start the pony, and how well it runs at low idle and under full power. If any of those arent' satisfactory, it's time for a tune-up.

Please send us an update as you work through it and get it back to easy-starting condition. We love pictures and videos too!

(Edit - apologies, forgot one of your questions. Yes many parts are still available new, and there are plenty second-hand. machinerytrader.com has a parts search link that you can put your part number into, and most parts can still be found. I try to buy parts from Cat themselves when the price is not horrendous because so long as the parts are still sold, they will still make them)