"Worry about something?"
"Worry for something?"
Which sentence is grammatically right?
Both are grammatically correct, and I've heard both used.
That said, I would lean toward using "Worry about something", because It's less vague. If I said, "I worry for John", it's possible (but not probable) that John has somehow hired you to worry.
If, on the other hand, I said that I worry about John, then there's virtually no way to misinterpret the sentence. The subject of your worrying is John.
You usually say that you "worry about" something, rather than "worry for" it. At least, if the "something" here is a thing that could cause trouble. You can worry about money, worry about your health, worry about the future of the country, etc.
When you say you "worry for" something you mean that you are concerned, not that thing will do harm, but that it will suffer harm. That is, if you say, "I worry for Fred", that means that you are concerned that something bad will happen to Fred. If you say, "I worry about Fred", that could mean that you think that something bad will happen to Fred, or it could mean that you think Fred will do harm to you or others.
So for example: "Bob is threatening Sally. I worry about Bob." You think Bob may harm Sally. "I worry for Sally." You think Sally may be harmed. You could also say, "I worry about Sally", hence "about" here is ambiguous.
You can also "worry for" a period of time, e.g. "I worried about that for six months".
I'm sure you could "worry for" other things that could follow the preposition "for". Like you could say, "I worry for $10 an hour", meaning if someone pays you $10 you will do their worrying for them for an hour.