Well, if you don't feel like putting in an entirely new floor (1" isn't
much worth it I suppose), I'd maybe just put a decorative trim around
it, almost like a door threshold. I'd buy a 2" wide piece of moulding
of some kind, like you would use around a door frame, but just lay it
flat on the floor. The round parts will be a pain, but you can do it
with some careful cutting, jigsawing, utility-knifing, etc.

Paint all surfaces of the trim after cutting, but before installing,
with something decently waterproof. I'd paint it to match the floor
(white, clear wood finish, whatever). Make sure the floor is absolutely
dry. Then run a little caulk all along the shower/wood crack, along the
top edge of the vinyl, lay down the trim, and fasten with a couple of
finish nails. THe idea is to keep water out from underneath the vinyl
especially, and out from under the trim as well.

If the curves are big, you might start with a 4" wide piece of trim,
lay them out, then "scribe" a line to match the curve of the shower,
then cut it down leaving only about 2" remaining.