You can go there by night bus from Yangon as I described in "The night bus experience".
Just plan to spend 2-3 days there which is just fine. I went there during high season so spending the whole day outside was impossible cause the temperature went almost until 40 degrees. I tried to wake up early in the morning to catch the sunrise by the temples where you can easily go by bicycle from Nyaung U to avoid the horrible midday heat. Then when coming back before 12 pm I just took a nap recapturing the rest of my daily sleep and woke up again for lunch time. Beeing completly alone in Nyaung U wasn't that easy even though I was traveling by myself. I never knew getting into a chat with other solo backpackers would be so easy during eating time which was very delightful. When sitting alone at a table there is always someone else sitting alone and automatically you start talking.
A must-to-do in Nyaung U is definitely to rent a bicycle and pass by the temples on your way to Old Bagan, either during sunrise or sunset, visit the village market and try some local food.
This is the perfect place if you want to get to a laid-back relaxing village