If you install the Play Services APK mentioned above, you also need a legacy version of the Play Store (v16.x or lower). The modern Play Store UI will crash on Android 4.1 even if the Services are working.
However, recognize the limitations. You are maintaining a legacy system. The Play Store may stop working entirely within the next year, and fewer apps will support the 4.1 API daily. Enjoy the nostalgia, but do not rely on Android 4.1 for critical daily tasks. Google Play Services Apk Android 4.1
If you are attempting this on Android 4.1, you must look for a specific code in the filename, usually starting with , which denotes compatibility with Android 4.1 (API 16). If you install the Play Services APK mentioned