By motorbike – You can hire self-drive motorbikes from around $15 per day at a number of places in downtown Mandalay, including Mandalay Motorcycle Rental – but it is a long trip, and almost a whole day (round trip) – heading south first to Sagaing and then back north along the western banks of the Irrawaddy to Mingun.
Having a phone with GPS is a very useful tool for this journey – if you do not have access to Myanmar mobile networks, make sure you pre-load the local area map when you have access to wifi.