Diagnosis
You are right to be careful. The screen has a delicate anti-glare coating. Standard cleaners will strip it.
How to Fix It
Use a dry, soft microfiber cloth. Gently wipe in one direction. For fingerprints, dampen a second microfiber cloth with distilled water only. Wring it out completely. Wipe gently, then immediately dry with the first cloth.
If That Didn't Work
For stubborn spots, use a 50/50 mix of distilled white vinegar and distilled water. Dampen the cloth, never spray the screen. Dry immediately.
Heads Up
Never use window cleaner, alcohol, ammonia, or abrasive cloths. Do not press hard. The coating cannot be replaced.