According to the way the software operates, the purchased music should automatically sync to the iPod. However, as we all know that iTunes can cause problems, it's possible for iTunes to miss the sync feature since the devices has been configured to work with another application (eg. TuneJack). I recommend a complete wipe (purchases will be safe!). Wipe the iPod then repeat the procedure with TuneJack and attempt to download the purchased music to the iPod while it's connected to PC.