Hello and welcome. Nice pickup for your first project.
Yes, the K181QS is correct and if it can be rebuilt parts are readily available.
If it needs replaced, sometimes not so easy depending on your area.
There would be nothing in particular that would be different about restoring an 800 than any other cub. But the fact that it is of the QuietLine series, your biggest concern will be with the rubber motor mount system (aka iso mounts). When they are shot, a lot of damage gets done because of engine movement.
Another area will be your clutch. The gear drive Quietlines use a flex disc as a clutch plate and when the iso mounts go bad and let the engine move around, the flex disc ends up cracking. If it all needs replaced you would have the option of using original type parts ($$$$$) or solid motor mounts and old style clutch parts.
You should get famiiair with the Tech Library, your gonna need these....
Chassis Svc. Man.
https://s3.us-east-2.amazonaws.com/c...ervice+Man.pdf
Engins Svc. Man
http://www.kohlerengines.com/engines...df/tp_2379.pdf
Q/L Op. Man.
https://www.onlycubcadets.net/pdf/QL%20Op%20Manual.pdf