Correct Sentences: Anything he does wouldn't surprise me. Correct and natural. Slightly formal, but fine. Whatever he does wouldn't surprise me. Correct. This is a common and natural expression. Nothing he does would surprise me. Also perfectly natural. Often used in speech. Slightly Off / Needs Correction: I won’t be surprised whatever he does. Not standard grammar. Better: “I won’t be surprised by whatever he does.” “Whatever he does, I won’t be surprised.” (You need to add “by” or rearrange the sentence.) I wouldn't be surprised whatever he does. Also not standard. Better: “I wouldn't be surprised by whatever he does.” “Whatever he does, I wouldn’t be surprised.”

Technically they're all valid, but with different levels of sophistication and different nuances of meaning.