NADYM, Russia—Beset by Western sanctions, Russian President Vladimir Putin is turning the national economy inward, enlisting local tycoons for the biggest infrastructure-investment program since Soviet times.

The government plans to spend the equivalent of $100 billion during the next five years on dozens of bridges, ports and transportation links, including the Northern Latitude Way railway, a revived train route through the frozen Arctic steppe first dug decades ago by gulag laborers.

The...