TOKYO—Japan’s services sector activity contracted at the fastest pace in five months in January in a sign businesses faced pressure from a record surge in new coronavirus infections due to the Omicron variant. The world’s third-largest economy has seen COVID-19 cases surge in recent weeks, forcing the government to roll out tougher curbs across much of the country in response to a rise in hospital admissions driven by Omicron. The final au Jibun Bank Japan Services Purchasing Managers’ Index (PMI) slumped to a seasonally adjusted 47.6 from the prior month’s 52.1 and a 48.8 flash reading. That marked the fastest decline in business activity since August, while outstanding business saw the sharpest rate of reduction in four months, the survey showed. Service firms cut staffing levels at the fastest pace in 20 months and became less optimistic that activity would rise over the next year, with positive sentiment reaching a …