First one, I think you could just say たんじょうびに and leave out のひ but I'm not sure. Maybe the equivalent in English is, "On the day of (my) birthday..." but again I'm not sure.
Second one, じゅうじはんにおきました。By the way, じゅうじはん is 10:30, not 10 which is just じゅうじ.