How to describe region around a white hole using a metric tensor? A white hole is a hypothetical region in spacetime predicted by general relativity. Although none is discovered, it exists on paper. With a metric tensor, it should be possible to simulate the movement of a test particle around it.
Can the Schwarzschild metric approximate space around a white hole? If yes, what parameter is different from a black hole metric? If not, what metric tensor can be used to describe this region?
Finally, does a white hole have a negative mass? What allows it to “repel” matter?
 A: 
Can the Schwarzschild metric approximate space around a white hole?

The Schwarzschild metric already describes the space around a (spherically symmetric, eternal) white hole. No parameter needs to be changed.
More specifically, its analytic continuation describes a white hole and a black hole both "in the same place". Inside the event horizon, spacetime is split into the white hole (in the past, with outflowing geodesics) and the black hole (in the future, with infalling geodesics).
For more general white holes, just time-reverse descriptions of black holes.

Finally, does a white hole have a negative mass? What allows it to “repel” matter?

A white hole doesn't repel matter, it attracts matter just like a black hole. Any material leaving it is slowed by its gravitational pull.
Intuitively, the singularity of a white hole could be viewed as "sourcing outflowing material at sufficient velocity to escape" (not a precise relativistic description), the time-reversed picture of how an object falls into a black hole with a trajectory that terminates at the singularity.
