Bikes are available everywhere and very cheap. It is absolutely a great way to get around in Mandalay.
Just make sure they give you a good one that won’t break down on you after the first turn. Let them put some more air in the tires as well. A lot of the attractions within the city are easily accessible by bicycle and the traffic is not too bad. You can spot a lot of interesting things along the way which makes it even more worthwhile.