Think you might have the wrong concept about repairs and enhancements to 50-60 year old motors. It's rarely a matter of getting part numbers and ordering parts -- exception would be maintenance parts like impellers and ignition parts (kind of remarkable if you think about it -- still readily available). For what you are after, though, you need to get more hands on -- need to find a donor motor, or even a complete '59 electric rumored to be in good running shape. You can probably get a complete motor for $200 or so -- less than you might pay for the parts you need. Check eBay, Craigs List, and local shops. Look for old stuff that nobody wants to work on (maybe came in as a trade). Shops often don't work on old motors, and they may go out to the recycler if you don't grab them.
If you do end up putting together a basic package, you need a bracket for your motor, a starter, a flywheel with ring gear (may already have that), an OUTBOARD solenoid, probably a starter button, wiring and battery. Wiring can be very simple.
Try the antique outboard site as mentioned -- post a free ad, asking for a '59 bracket for the 18 and a starter. See what you get. You might also show an interest in buying a complete electric-start motor or parts motor.