A wormhole solution necessarily violates the averaged null energy condition (ANEC). The positive energy theorem says that if the dominant energy condition (DEC) holds, then the ADM mass is strictly greater than zero for a non-flat spacetime. The DEC is stronger than the ANEC. So a wormhole must violate the DEC, and therefore the positive energy theorem doesn't apply, and there seems to be no conclusive answer as to the sign of the wormhole's ADM mass. It's of course possible that there is some other argument that I'm not aware of that would give a more definite answer about the sign of the mass.
Kip Thorne's group at Cal Tech did a lot of work on wormholes and CTCs in the 1980's. Some of their papers, as far as I recall, seemed to implicitly assume a positive ADM mass. They describe a mechanism by which any wormhole can always be made into an eternal time machine. This is briefly described, e.g., in Echeverria 1991, and their mechanism basically involves accelerating one mouth so that it has some cumulative time dilation. Reading those arguments, my understanding was always that that they were talking about manipulating the mouth by attracting it gravitationally with an external positive mass. But of course if the mouth had a negative mass, you could manipulate it just as easily using the repulsion of a positive mass. It might be helpful to comb through some of those old papers and see if they ever explicitly give some reason why the ADM mass should have a particular sign.
Echeverria 1991 - Echeverria, 1991, "Billiard balls in wormhole spacetimes with closed timelike curves: Classical theory," http://authors.library.caltech.edu/6469/