Yes, Medicare Part A will pay part of your hospital bills even if you also have work-related insurance. Which coverage pays first -- Medicare Part A or your private, employer-sponsored
insurance -- and how much each pays depends on the size of your employer. If your employer has 20 or more employees, then your employer-based health plan is the primary payer, with that insurance paying first and Medicare Part A paying some of the remaining costs. If your employer has fewer than 20 employees, Medicare would be the primary payer of your hospital bills, in which case your private, employer-sponsored health insurance would pay some of the costs that Medicare doesn't pay (like the Medicare Part A hospital deductible and copayments).
If and when you register for a hospital stay, give the hospital registrar your Medicare card and also the information about your employer-sponsored insurance. The hospital will then automatically submit your bills to both, and Medicare and the private insurance company will sort out between them which insurance pays how much of the bills. To read more about how Medicare and private insurance pay in these situations, you can look at Medicare's online booklet Medicare and Other Health Benefits: Your Guide to Who Pays First.