In the outskirts of Las Vegas, nestled between a low-rent office park and an AutoZone, construction crews are fitting together a new apartment building that a prominent roster of tech investors is betting will be the future of real-estate construction.

If all goes according to plan, the three-story building designed and built by Silicon Valley startup Katerra Inc. will rise from its concrete floor to completion in 90 days. The rapid pace is possible because custom panels of walls and cabinets are built by an Arizona assembly...