If the content is "SD" (standard defintion), you should be able to sync it to an iPod nano 5th gen. It can generally play videos up to 640 by 480 pixels in resolution. "HD" content is beyond that resolution, so iTunes will not sync it to the iPod.
If you use automatic syncing, you need to set up the iPod's iTunes U tab in iTunes. Select the iPod in iTunes. There is a row of "tabs" (starting with Summary). Click on iTunes U there. This tab is where you tell iTunes how to sync iTunes U content. Sync iTunes U needs to be checked. Below that, select the items you want to sync to the iPod.
NOTE: When you Apply the change, the selections from the iTunes tab and the other tabs (Music, Podcasts, etc.) sync to the iPod, replacing its current content. If you currently use the manual method to load the iPod, do not do the above. Instead...
If you use the manual method to load the iPod (drag and drop items to the iPod in iTunes), you can do the same with the iTunes U content. Select iTunes U under LIBRARY in the iTunes sidebar, and drag the items you want the iPod.
It's not an "app." On the iPod nano, iTunes U just appears as another selection on the screen menu. However, it wll not appear on the screen menu, if you currently do not have any iTunes U items loaded on the iPod. If you load some items as explained in my previous reply, you should see iTunes U (below Podcasts) on the iPod's main screen menu.