Learn Dutch: Hij strijkt het overhemd pas vanavond. - He is not ironing the shirt until tonight.

This is about middle field word order. Because het overhemd is a definite direct object (we know exactly which shirt it is), it must be placed before adverbs of time. Therefore, it comes before the time phrase pas vanavond.

What would happen to the word order if he were ironing "a shirt" (indefinite) instead of "the shirt"?

If the object were indefinite (een overhemd), it would move after the time expression. The sentence would then become: Hij strijkt pas vanavond een overhemd.

The English translation says "not until", but I don't see a negative word like niet in the Dutch sentence. How does that work?

The word pas means "only" or "just" when referring to time. Literally, this sentence says "He irons the shirt only tonight." In English, we usually express this idea using "not until". You don't need niet because pas already implies that the action won't happen before that specific time.

Why does the Dutch sentence use the present tense (strijkt) for something happening later tonight?

Dutch frequently uses the present tense to talk about the future, as long as there is a time word (like vanavond) that makes it clear the action is happening later. You could use the future tense with zullen, but using the present tense is much more common and natural here.