is the word "our" correct?

Yes, I would use "our" in this sentence, but you need to make screen plural.  There are also punctuation problems.

According to a recent report, reality TV now accounts for 57% of all programmes found on our screens.

A sentence always begins with a capital letter. It is a good idea to practice writing correctly even when you think it doesn't matter.  It always matters. Lazy writing can lead to missed jobs and promotions.  Good writing habits will remove that risk.