Hi Helen,
Yes, you can get a tourist transit boat from Bagan to Mandalay. I did it last year. It was arranged via a travel agent so I can't tell you the specific cost as it came in a package of travel and lodging.
The journey took about 11-12 hours as it is against the current. Going this direction the boat will have fewer passengers as it takes significantly longer due to the current. In fact two of us had the entire boat to ourselves.
The boat left Bagan at day break and arrived in Mandalay during the afternoon "rush hour" which was a nice contrast to Bagan. The boat we were on had a TV/DVD player to pass the time but the scenery is interesting for much of the time. The seats were coach style.
It is a great place. Have fun!