There are plenty of horse carts for hire in Nyaung U. I didn't ride one here but they would make for a pleasant ride to the Jetty or possibly all the way to Old Pagan. It is probably better to make sure of the price before you get in. So you don't have any problems when it is time to pay. Horse carts are more expensive than the rickshaws as 2 people have to be fed..The driver and the horse!
I didn't have the pleasure of riding on this bus but there are plenty of shorter journeys done on these kinds of buses. For sure you will be sitting next the locals and their livestock! Chickens and goats are very common! I've riden on plenty of Asian buses with chickens and goats! LOL The first time you see it you might frown but take it in stride and the chickens will be your friend after sitting next to her for 6 hours!!