He went to the cinema if a good movie was "on". He went to the theater if a good play was "on" or "up"?

'On' sounds perfectly right to my ears. I've never heard anyone say "there's a play up" (At least, not while talking about theatre. It might be used in discussing sports)

Does the phrase 'putting on a show' sound OK to you? Then you should understand about a show being 'on'.

Here are a few examples in the wild.

https://almeida.co.uk/between-a-new-play-and-an-adaptation

I'm interested in the way you talk about that I think is something kind of unusual, or sort of relatively unusual about you for writers, at least over here the prevailing culture is that the writers think about the text. And one of the things I'm always mindful of when you send us a new play, or there's a play on that I come and see when I'm over, is that your work really does think about all the different elements.

http://www.stagereviews.co.uk/londonwest-end.html

There’s a play on at the National Theatre at the moment called The Welkin, about a woman in 18th century England who appeals for clemency from execution by claiming she’s pregnant – pleading the belly, as it was known.

https://www.gazette-news.co.uk/news/17973339.paul-t-davies-column-radiant-vermin-headgate-theatre-colchester/

But if you are prepared to take a risk, there’s a play on this week at the Headgate Theatre in Colchester which is intriguing in content and performance, and which not even I have heard of.