In this sentence, 'hat' is the subject. It's literally 'A blue hat is near me' (which is one of the ways possession is indicated in Hindi), so you have to conjugate the sentence with hat as the subject, not 'me'. Hence 'hai', not 'hoon'.
As I am reaching on next level This app is started irritating day by day. Now in this the right answer is as per app मेरे पास एक नीली टोपी है। and I wrote मेरे पास एक नीली टोपी है and it says that your answer is wrong.