We study algebra endomorphisms and derivations of some localized down-up algebras A S (r + s, −rs). First, we determine all the algebra endomorphisms of A S (r + s, −rs) under some conditions on r and s. We show that each algebra endomorphism of A S (r + s, −rs) is an algebra automorphism if r m s n = 1 implies m = n = 0. When r = s −1 = q is not a root of unity, we give a criterion for an algebra endomorphism of A S (r + s, −rs) to be an algebra automorphism. In either case, we are able to determine the algebra automorphism group for A S (r + s, −rs). We also show that each surjective algebra endomorphism of the down-up algebra A(r +s, −rs) is an algebra automorphism in either case. Second, we determine all the derivations of A S (r + s, −rs) and calculate its first degree Hochschild cohomology group.