Take a pick-up van from those that regularly pass through Mandalay, or hire a taxi, rickshaw or bicycle. The bike ride can take between 45 minutes to 1 hour. Another alternative is to sign up for a tour of the Amarapura region, but remember that you may not have the time and freedom you wanted.
Myanmar's very low prices allow you to take successive pick-up trucks between towns or monuments or have a taxi, rickshaw or bicycle at your disposal without any financial damage worthy of mention. Once the vehicle is sorted out, departing early from Mandalay, you can explore the ancient cities around Mandalay – Amarapura, Inwa etc and their monuments with relative ease.
Admission to the Mandalay Archeological Zone costs 10 dollars (7.5€). Includes Inwa, Amarapura and the visit to the u-Bein bridge. The boat trip on Lake Taungthaman should cost around €2.