"What do you do in school?"

[Ju]

So it means to hand, at hand, on hand and handy are all grammatical. Is it right?

 

[Matthew Wai]

I think you are right.

 

[Rover_KE]

... but don't get the idea that they're interchangeable in all contexts.

 

[Charlie Bernstein]

I would use to hand there.

I'm surprised! In American English, "on hand" is correct, grammatical, and natural. "To hand" would be wrong unless you say whom it's handed to:

- I didn't have the book to hand to you.
- I didn't have a book to hand Robert. (The word to is understood.)

 

[Charlie Bernstein]

So it means to hand, at hand, on hand and handy are all grammatical. Is it right?

It depends. They have different meanings and uses.