Hi Finn. It’s great to see a junior member involved in this hobby and in this forum. This is a rewarding hobby, and you will make some lifetime friends!
If this is like some other OMC motors from the 1940’s, the rubber that you see is a shock absorbing ring. There is an aluminum and bronze hub inside of it, and all of it is removable. Often times the rubber ring is reusable, but if not, there are likely new old stock parts in existence.
I have been able to “patch up” some of these rubber rings by removing them, cleaning all the grease off of everything, and using Permatex Ultra Black on the outside of the hub, and inside of the prop prior to reassembly. Your prop looks to be in good condition, so go with it.
Regarding gearcase grease, John Deere Corn Head Grease is the only product that I know of left on the market.
A club member had one of these running at a recent meet, and it ran nice and smooth!