Where can I print the details of Shenyang medical insurance card?

There will be a social security center under the Human Resources and Social Security Bureau, with many windows. You can print it in the medical insurance window and bring your ID card.

The details of the payment of the old-age insurance you printed can only prove that it was paid by the old-age insurance. Some areas don't have to pay all the insurance, so it doesn't prove that you have medical insurance.

Further reading: How to buy insurance, which is good, and teach you how to avoid these "pits" of insurance.