I notice: The identity function f(x) = x is order-preserving in all three cases. I think this is unique.

Multiplication by a positive constant f(x) = c*x is addition-preserving and order-preserving, but not metric preserving.
Annihilation f(x) = 0 is addition-preserving and order-preserving, but not metric preserving.

Rotation through 180° f(x) = -x is metric preserving and addition-preserving, but not order-preserving.

Also, it looks like the addition-preserving endomorphisms are fully determined by the value of f(1).