Learn Italian: Lui non è stanco oggi.

You do not need to change the word order at all! To ask "Is he not tired today?", you just say the exact same words—Lui non è stanco oggi?—but make the pitch of your voice go up at the end of the sentence, just like you would in English when asking "He's not tired?".

I thought we usually drop words like lui (he) in Italian. Why is it used here?

You are right that subject pronouns are usually dropped! Including lui here puts special emphasis on the person. The speaker is specifically pointing out that he is not tired, perhaps to contrast him with someone else who is.

Why does non come before the verb è?

In Italian, the negative word non always goes directly before the verb it is negating. While in English we say "is not", Italian literally says "not is" (non è).

What if I wanted to say "She is not tired today"?

You would need to change two things. First, change lui (he) to lei (she). Second, because Italian adjectives change their endings to match the gender of the person they describe, stanco changes to the feminine form stanca. The sentence becomes: Lei non è stanca oggi.