Your mower should come with a 20 oz bottle of oil for the initial fill. Use this oil for 5 hours, then drain, and put in the 5W- or 10W30 synthetic of your choice. These Honda 160's are easy on oil.
My experience with this engine goes back to when I purchased a mower equipped with this engine back in 2000. I have kept an hour meter on this engine, and it has just passed 500 hours. After the original fill, which I dumped at about 5 hours, I used Mobil 1 for the next few years. It was changed once a year, at around 60 hours. It has never required oil to be added during the OCI. It always stays right on the full mark. I used 5W30 up until last year, when I used up my old supply of SL rated M1. I then began to use M1 High Mileage oil, 10W30, which is still SL rated. I have noted that the Amsoil 10W30/30 oil recommended for small engines is still SL rated, as is the 5W30 oil Briggs & Stratton sells under their own name. German GC 0W30 was also still SL rated.
I'm sure most any SM oil would be fine; I just feel more comfortable with the SL oils and their non-catalytic converter saving formula.