If you are staying in Nyaung U taking the Horse cart to Old Bagan could be quite fun. I'm sure you could hire a cart for the whole day. It might not be the most comfortable way to travel for a day but it certainly would be interesting!
Horse carts were used by alot of locals to get around. I'm assuming that they are quite inexpensive. Because the locals used them quite a bit. You will have to establish where you want to go and the price before you get in.I did'nt ride in any in Bagan because we had our bikes. One day while we were riding our bikes my son found a horse shoe in the road and he brought that home. A cheap souvenir!
I would have to say that the bicycle is the best way to say the area around Bagan. (as long as your fit enough) We rented bicycles for US$.55cents per day so they are quite affordable and there are many places renting them! If you are there it would be virtually impossible to walk and see the sites as they are quite spread out. We stopped under the shade of this tree to take a look at our map and figure out where we were going.
I don't think the horse cart drivers keep really busy. As most of the time they were just lounging and sleeping while waiting for a customer.
My son got lucky as we where able to find a slightly smaller size bike to rent for him. The owner of our guest house had a younger brother so we where able to rent his.
Many local people use bicycles to get around. There are a few brands to chose from. I know there are some from India and some from China and I think a couple of local brands too.