We stayed in Lesser Town for three nights before our tour, and had just spent 5 nights in Vienna. When we checked in to our B&B and asked about a laundromat, we were told there weren't a lot of them in Prague. Our B&B offered to do our laundry for us, and they did it quickly and for a very reasonable rate. The sheer convenience made it worthwhile; but the fact that it was done with such care made it even better.
You might want to contact your hotel ahead of time to find out their suggestions about options for doing laundry.
Which tour are you doing? On our Best of Eastern Europe tour, there was apparently an option to have laundry done in Krakow. We didn't look into it, because we didn't need laundry done at the time. After that, you may need to wait until Budapest and will need to drop it off yourself.